#72319613

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 10895 conjugacy classes of subgroups; only the 29653 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

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