#72330784

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄³

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 9277 conjugacy classes of subgroups; only the 1145 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,d^2=ghi,e^2=fhij,b^a=bi,c^a=cj,c^b=cg,d^a=dfghi,d^b=dij,d^c=dj,e^a=efgi,e^b=efij,e^d=eghj

Conjugacy classes & character table

124 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.8);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
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.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);