#72250108

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

62093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=i,d^2=j,b^a=bh,c^a=cg,d^a=dij,d^b=dj,e^c=ej,f^c=fi,f^d=fj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250116, #72297794
maximal normal subgroup of	#72250116, #72297794
maximal quotient of	#72250311, #72298568, #72298577

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