#72285153

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216457
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20586 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14)(4,11)(5,23,27,10)(6,28,32,12)(7,13)(8,17)(9,22)(15,35,47,43)(16,44,24,57)(18,45,29,33)(19,37)(20,38,25,40)(21,39,30,42)(26,41)(31,56)(34,51,55,49)(36,52)(46,63)(48,60)(50,59)(53,64)(54,62)(58,61);(1,3,4,17)(2,8,11,14)(5,24,25,51,27,16,20,49)(6,29,30,47,32,18,21,15)(7,19,22,52)(9,37,13,36)(10,34,40,44,23,55,38,57)(12,35,42,45,28,43,39,33)(26,53,54,46)(31,50,58,48)(41,63,62,64)(56,60,61,59);(1,4)(2,9,13,11)(3,15)(5,20,27,25)(6,30)(7,22)(8,33)(10,38)(12,42,28,39)(14,43,37,35)(16,48,24,46)(17,18,52,29)(19,47)(21,32)(23,40)(26,54,31,58)(34,59,55,64)(36,45)(41,61)(44,63)(49,53)(50,51)(56,62)(57,60);(1,5)(2,10)(3,16)(4,20)(6,31)(7,27)(8,34)(9,38)(11,40)(12,41)(13,23)(14,44)(15,46)(17,51)(18,53)(19,24)(21,54)(22,25)(26,32)(28,56)(29,50)(30,58)(33,59)(35,60)(36,55)(37,57)(39,62)(42,61)(43,63)(45,64)(47,48)(49,52)

Polycyclic presentation:

2^11|a^2=g,b^2=e,c^2=ghjk,e^2=g,f^2=gijk,b^a=be,c^a=chijk,c^b=cf,d^a=dh,d^b=dhik,d^c=dhij,e^a=eg,e^c=eh,e^d=eijk,f^a=fhj,f^b=fghijk,f^c=fgjk,f^d=fijk,f^e=fij,g^c=gjk,g^d=gk,g^f=gk,h^a=hk,h^b=hi,h^c=hjk,h^e=hjk,h^f=hj,i^a=ij,i^b=ijk,i^c=ij,i^e=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);