#72262483

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

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 25826 conjugacy classes of subgroups; only the 31825 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 20 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,f^2=j,b^a=bi,c^a=ci,e^c=eh,e^d=eij,f^a=fi,f^d=fj,g^b=gi,g^c=gj,g^f=gj

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262626
maximal normal subgroup of	#72262626

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