#72330728

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×#15257	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7054 conjugacy classes of subgroups; only the 1159 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=fhj,e^2=h,b^a=bf,c^a=cfhj,d^a=dhi,d^b=dhi,e^b=efg,e^c=efg,e^d=eh

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331189, #72331237, #72331612
maximal normal subgroup of	#72331189, #72331237, #72331612

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