#72249476

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 38944 conjugacy classes of subgroups; only the 29395 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)(17,18,24,26)(19,28,31,22)(20,25,30,32)(21,23,27,29);(3,7)(6,8)(17,19,24,31)(18,22,26,28)(20,29,30,23)(21,25,27,32);(1,2)(3,8)(4,5)(6,7)(11,15)(14,16)(17,20)(18,25)(19,29)(21,28)(22,27)(23,31)(24,30)(26,32);(3,7)(6,8)(9,10)(11,16)(12,13)(14,15)(18,26)(21,27)(22,28)(25,32);(2,5)(6,8)(10,13)(14,16)(17,21)(18,23)(19,25)(20,22)(24,27)(26,29)(28,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,22)(18,19)(20,21)(23,25)(24,28)(26,31)(27,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,23)(18,27)(19,30)(20,31)(21,26)(22,25)(24,29)(28,32)

Polycyclic presentation:

2^10|a^2=i,b^2=i,b^a=bi,d^a=di,e^a=eh,e^c=ehj,e^d=ehi,f^b=fhj,f^c=fj,f^d=fj,g^b=gi,g^e=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, 4, 1, F.4*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.8*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, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);