#72332499

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂²×#2613	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

31383 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=h,b^a=bg,c^a=ci,c^b=ch,d^a=dgj,d^b=df,d^c=dhi,e^c=efh,e^d=efh

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333201, #72333256, #72333332
maximal normal subgroup of	#72333201, #72333256, #72333332

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