#72326878

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 7506 conjugacy classes of subgroups; only the 4245 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=hi,d^2=h,f^2=ghij,b^a=bi,c^a=cj,c^b=cg,d^b=di,d^c=dj,e^a=eh,e^b=ehi,e^c=ej,e^d=eh,f^a=fgi,f^b=fj,f^c=fj,f^d=fi,f^e=fghi

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326901, #72328045
maximal normal subgroup of	#72326901, #72328045
maximal quotient of	#72335784, #72335794, #72336493

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