#375012

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282208);