#72311673

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 7995 conjugacy classes of subgroups; only the 29441 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

(1,2)(3,7)(4,5)(6,8)(11,16)(13,14)(17,18)(19,22)(20,23)(21,24);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23);(9,10)(11,14)(12,15)(13,16)(19,23)(20,22);(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24);(2,5)(6,7)(9,10,12,15)(11,14,16,13)(17,20)(18,23)(19,24)(21,22);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,15)(14,16)

Polycyclic presentation:

2^10|e^2=j,c^b=ch,d^a=dhj,d^b=dj,d^c=dhj,e^a=ei,e^b=ej,e^c=ehj,e^d=ehj,f^a=fij,g^e=gij,g^f=gj

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311697, #72342349
maximal normal subgroup of	#72311697, #72342349

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