#72249502

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

At least 20957 conjugacy classes of subgroups; only the 29417 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,b^a=bi,e^a=eh,e^c=ehj,e^d=ei,f^b=fhj,f^c=fj,g^a=gi,g^b=gi,g^e=ghj,g^f=gj

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567
maximal normal subgroup of	#72249567

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