#72256686

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

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

117734 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|c^2=j,d^2=hi,e^2=ij,c^a=cj,d^a=di,d^b=dh,d^c=dh,e^a=ei,e^b=ei,e^c=ei,e^d=ei,f^b=fj,f^c=fj,f^d=fi,f^e=fi,g^e=gj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256702
maximal normal subgroup of	#72256702

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