#72241154

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 8966 conjugacy classes of subgroups; only the 31873 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:

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

Polycyclic presentation:

2^10|f^2=j,b^a=bi,f^a=fh,f^b=fhj,f^c=fj,f^e=fhj,g^a=gj,g^b=gi,g^d=gi,g^f=ghij

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241176, #72267417
maximal normal subgroup of	#72241176, #72267417
maximal quotient of	#72328076, #72328461, #72329599, …

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