#72331709

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 11318 conjugacy classes of subgroups; only the 3969 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 20 points:

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

Polycyclic presentation:

2^10|c^2=i,d^2=hi,e^2=g,b^a=bj,d^b=dg,d^c=dh,e^b=eg,e^c=egi,e^d=eghi,f^a=fi,f^c=fi,f^d=fgh

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331723
maximal normal subgroup of	#72331723
maximal quotient of	#72333338

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