#411202

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=gi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi

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, 318398);