#72284072

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

3567 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)(4,18)(5,23)(6,27)(8,30)(14,43)(15,40)(16,49)(17,42)(20,47)(21,25)(26,29)(31,57)(32,36)(33,58)(34,53)(37,55)(39,45)(41,51)(44,62)(46,61)(48,63)(50,64)(54,56);(1,2)(3,14,17,45)(4,19,18,9)(5,24,23,10)(6,28,27,11)(7,12)(8,16)(13,39,42,43)(15,46,20,44)(21,34)(22,38)(25,53)(26,55)(29,37)(30,49)(31,48)(32,51)(33,50)(35,52)(36,41)(40,62,47,61)(54,60,56,59)(57,63)(58,64);(1,3,7,17)(2,8)(4,20)(5,13)(6,16,27,51)(9,33,19,31)(10,36,24,30)(11,14)(12,32)(15,18)(21,47,25,40)(22,50)(23,42)(26,41)(28,45)(29,49)(34,58)(35,46,38,44)(37,43,55,39)(48,52)(53,57)(54,64,56,63)(59,61)(60,62);(1,4)(2,9)(3,15)(5,25,23,21)(6,22,27,52)(7,18)(8,31)(10,34,24,53)(11,38,28,35)(12,19)(13,40,42,47)(14,44,45,46)(16,50,51,48)(17,20)(26,54)(29,56)(30,57,36,58)(32,33)(37,59)(39,61)(41,63)(43,62)(49,64)(55,60)

Polycyclic presentation:

2^11|b^2=ghj,c^2=ik,d^2=hik,e^2=h,f^2=k,b^a=bghj,c^a=ce,c^b=cf,d^a=dg,d^b=dgk,d^c=dhijk,e^a=eh,e^b=eg,e^c=ehj,e^d=eik,f^a=fg,f^b=fi,f^d=fijk,f^e=fij,g^b=gjk,g^c=gi,g^d=gj,g^e=gjk,h^b=hjk,h^d=hk,h^f=hk,i^a=ij,i^b=ik,i^d=ik,i^e=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285547, #72340332, #72340399, …
maximal normal subgroup of	#72285547, #72340332, #72340399, …

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