#72252235

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

58321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 32 points:

(18,22)(23,26)(24,29)(30,32);(9,10)(11,13)(12,14)(15,16)(19,25)(23,26)(27,31)(30,32);(9,11)(10,13)(12,16)(14,15)(20,28)(24,29)(27,31)(30,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,22)(19,25)(24,29)(27,31);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^10|e^2=h,d^c=dh,e^a=ei,e^b=eh,e^d=ej,f^b=fi,f^d=fj,f^e=fi,g^c=gi,g^e=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252242, #72283114, #72340406
maximal normal subgroup of	#72252242, #72283114, #72340406
maximal quotient of	#72283161, #72333429, #72334064, …

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