#72327643

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 39855 conjugacy classes of subgroups; only the 29389 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|f^2=hj,g^2=j,b^a=bi,c^b=ch,f^a=fj,f^c=fj,f^d=fj,f^e=fhj,g^a=ghj,g^c=gj,g^d=gi,g^e=gj,g^f=gij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72327934, #72328649, #72329738

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