#72319702

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 38768 conjugacy classes of subgroups; only the 29341 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

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