#72304331

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 12353 conjugacy classes of subgroups; only the 29713 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=j,c^2=ij,d^2=hij,g^2=j,c^a=ci,d^a=dij,d^b=dij,d^c=dj,e^a=ei,e^b=ei,e^c=ei,e^d=ei,g^e=gj,g^f=ghj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304523
maximal normal subgroup of	#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, 1, F.8);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);