#72330792

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

27565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,24)(22,23)(25,26)(27,32)(28,30)(29,31);(9,10)(11,15)(12,13)(14,16)(17,19,22,20)(18,21,23,24)(25,27,30,29)(26,31,28,32);(10,13)(14,15)(18,23)(19,20)(26,28)(31,32);(2,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,20,22,19)(18,21,23,24)(25,28,30,26)(27,32,29,31);(1,2)(3,4)(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,21)(18,19)(20,23)(22,24)(25,29,30,27)(26,32,28,31)

Polycyclic presentation:

2^10|a^2=f,b^2=fgh,d^2=gh,e^2=fhj,b^a=bg,c^a=cj,c^b=cgj,d^a=dfgh,d^c=dj,e^a=efgi,e^b=egj,e^d=eghij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

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