#72329192

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 8426 conjugacy classes of subgroups; only the 975 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,d^2=hij,e^2=g,b^a=bg,c^a=cj,c^b=cf,d^a=dhi,d^b=dij,d^c=dhij,e^a=efhi,e^b=efghj,e^c=efhij,e^d=efhj

Conjugacy classes & character table

118 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.10);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7);;
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.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);