#72335037

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂³	C₂³×C₄²
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

64577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=f,d^2=f,b^a=bh,c^a=chi,c^b=cg,d^a=di,d^c=dghi,e^b=egi,e^d=egi

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335048, #72335051, #72335055, …
maximal normal subgroup of	#72335048, #72335051, #72335055, …
maximal quotient of	#72335083, #72337406

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