#72322202

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₈²	C₂⁵
socle	C₂²	C₂²×#1580
prime radical	#7623228	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×#1312 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

3406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|b^2=f,e^2=gi,f^2=h,g^2=ik,h^2=j,i^2=k,b^a=bg,c^a=cj,d^a=dk,e^a=eg,e^b=ei,f^a=fi,f^e=fk,g^a=gi,g^b=gk,h^a=hk,i^a=ik

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322318, #72322953, #72323108
maximal normal subgroup of	#72322318, #72322953, #72323108

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