#72328259

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 11586 conjugacy classes of subgroups; only the 3961 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=gi,d^2=gj,f^2=gij,b^a=bh,c^a=cg,c^b=cghi,d^a=dgij,d^c=dij,e^a=ei,e^c=egi,e^d=egj,f^a=fgij,f^c=fhj,f^e=fgij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328397, #72336751
maximal normal subgroup of	#72328397, #72336751

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