#72331001

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7058 conjugacy classes of subgroups; only the 1075 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(6,7)(10,12)(13,14,17,18)(15,19,20,16)(21,22,25,27)(23,26,28,24);(5,6)(7,8)(9,10)(11,12)(21,23,25,28)(22,26,27,24);(6,7)(14,18)(16,19)(22,27)(24,26);(2,4)(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,28)(25,27);(1,2)(3,4)(5,7,8,6)(13,16,17,19)(14,20,18,15)(21,24,25,26)(22,28,27,23)

Polycyclic presentation:

2^10|a^2=h,b^2=f,e^2=ij,b^a=bhi,c^a=ch,c^b=chij,d^a=df,d^c=df,e^a=eij,e^b=ehij,e^c=eij,e^d=eg

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331105, #72331129, #72331639
maximal normal subgroup of	#72331105, #72331129, #72331639

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