#72225476

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
socle	C₂	D₄×#1580
prime radical	#7623661	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #493216 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,c^2=f,d^2=gj,e^2=hjk,f^2=i,g^2=jk,h^2=k,j^2=k,c^a=ch,c^b=cg,d^b=dg,d^c=dj,e^a=eh,e^b=ej,f^a=fk,f^b=fj,f^d=fk,g^b=gj,g^c=gk,h^a=hk,i^b=ik,j^b=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225512, #72227158, #72227171
maximal normal subgroup of	#72225512, #72227158, #72227171

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