#72237366

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	C₂²×C₄	C₂⁶
commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 13115 conjugacy classes of subgroups; only the 32077 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,b^a=bi,c^a=ci,c^b=ci,f^d=fj,f^e=fh,g^d=gi,g^e=ghj

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237405, #72264089, C₂×#72237366
maximal normal subgroup of	#72237405, #72264089, C₂×#72237366
maximal quotient of	C₂×#72237366

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