#72234883

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 7951 conjugacy classes of subgroups; only the 29725 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=h,b^a=bh,c^b=ch,d^a=dh,d^c=dhj,e^c=ei,e^d=ei,f^c=fj,f^d=fj,g^e=gij,g^f=ghij

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234898, #72304523
maximal normal subgroup of	#72234898, #72304523

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