#72267375

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 13588 conjugacy classes of subgroups; only the 31881 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:

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

Polycyclic presentation:

2^10|b^2=h,e^2=h,b^a=bi,d^b=di,e^d=ei,f^b=fh,f^e=fh,g^a=gi,g^b=gj,g^c=gj,g^e=ghi

Conjugacy classes & character table

196 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, 2, F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);