#387225

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,24)(22,23);(1,4,5,8)(2,6,3,7)(10,14)(13,15)(19,24)(21,23)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(2,3)(6,7)(11,16)(13,15)(18,22)(19,24)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=efg,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294421);