#72321391

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄²	C₂⁴
commutator	C₂³	C₂²×C₄×C₈
socle	C₂⁴	C₂²×C₄²
prime radical	C₂³×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

4521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,d^2=f,e^2=i,h^2=j,b^a=bj,d^a=dg,d^b=df,d^c=df

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321430, #72321510, #72322207
maximal normal subgroup of	#72321430, #72321510, #72322207
maximal quotient of	#72322701

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