#72286031

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
prime radical	#72213523	C₂
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

3628 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:

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

Polycyclic presentation:

2^11|b^2=gjk,f^2=hik,c^a=ce,c^b=cf,d^a=di,d^b=dghi,d^c=di,e^b=eg,f^a=fgij,f^b=fh,f^c=fhik,f^d=fik,f^e=fij,g^c=gi,g^d=gk,g^f=gjk,h^a=hj,h^b=hjk,h^d=hk,h^e=hk,h^f=hk,i^b=ij,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340482, #72340497
maximal normal subgroup of	#72340482, #72340497

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, 2, F.7*F.10*F.11);;
gap> SetPower(coll, 6, F.8*F.9*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);