#366061

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, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^c=di

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, 273257);