#72267397

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

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 6118 conjugacy classes of subgroups; only the 32041 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)(13,14)(15,17)(16,19)(18,20);(5,6)(7,9)(8,11)(10,12);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,14,17,15)(16,19,20,18);(1,2,4,3)(7,12)(9,10)(13,14,17,15)(16,19,20,18);(14,15)(16,20);(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,15,17,14)(16,18,20,19);(1,3,4,2)(5,7)(6,10)(8,12)(9,11)(13,16,17,20)(14,18,15,19)

Polycyclic presentation:

2^10|c^2=j,d^2=j,f^2=j,g^2=j,c^a=cj,d^a=dj,d^c=di,e^a=eh,e^c=eh,e^d=eh,f^a=fj,f^e=fh,g^a=gj,g^b=gi,g^c=gh,g^d=ghi,g^e=gh,g^f=ghi

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