#72304372

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 12949 conjugacy classes of subgroups; only the 29701 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=i,d^2=hij,f^2=ij,g^2=j,c^a=ci,c^b=cj,d^a=di,d^b=di,d^c=dj,e^c=eij,e^d=eij,f^a=fi,f^b=fi,f^c=fj,f^e=fij,g^e=gj,g^f=gh

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