#72267390

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 16597 conjugacy classes of subgroups; only the 31869 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|e^2=h,f^2=j,d^c=di,e^a=ej,f^a=fj,f^e=fhj,g^b=gi,g^c=ghj,g^d=ghij,g^f=ghij

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267417
maximal normal subgroup of	#72267417

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