#72232072

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 6292 conjugacy classes of subgroups; only the 31937 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:

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

Polycyclic presentation:

2^10|g^2=j,c^b=ch,f^b=fj,f^c=fj,f^d=fj,f^e=fi,g^a=gj,g^d=gj,g^e=ghi

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234468, #72241335, #72241455, …
maximal normal subgroup of	#72234468, #72241335, #72241455, …
maximal quotient of	#72232191, #72232274, #72232573, …

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