#72340002

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68869 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=j,d^2=ghij,b^a=bh,c^b=ci,d^b=dij,d^c=dij,e^b=eh,e^d=eh,f^b=fhj,f^d=fhj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340005
maximal normal subgroup of	#72340005
maximal quotient of	#72340027, #72340046

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