#72321171

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	C₂×C₄	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₈²	C₂⁵
socle	C₂²	C₂²×#1580
prime radical	#7623249	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ #479952 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

3260 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)(11,19)(12,17)(13,18)(14,22)(15,20)(16,21)(23,28)(24,29)(25,27)(26,30)(31,32)(34,42)(35,47)(36,53)(37,44)(38,54)(39,40)(43,58)(45,57)(46,56)(48,55)(49,62)(50,59)(60,63)(61,64);(1,3,4,12,5,13,14,25,6,11,15,23,16,24,26,31)(2,7,9,17,10,18,22,27,8,19,20,28,21,29,30,32)(33,34,37,42,39,45,54,56,41,36,38,53,40,46,44,57)(35,48,49,58,51,61,62,64,47,43,50,55,52,60,59,63);(2,8)(3,11)(9,20)(10,21)(12,23)(13,24)(22,30)(25,31)(33,35)(34,43)(36,48)(37,49)(38,50)(39,51)(40,52)(41,47)(42,55)(44,59)(45,60)(46,61)(53,58)(54,62)(56,63)(57,64);(35,47)(43,48)(49,50)(51,52)(55,58)(59,62)(60,61)(63,64);(33,36,54,56,40,45,37,42,41,34,44,57,39,46,38,53)(35,48,62,63,52,60,49,55,47,43,59,64,51,61,50,58)

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,c^b=cj,d^c=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

236 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322318, #72323133, #72323147
maximal normal subgroup of	#72322318, #72323133, #72323147

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, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 3, 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);