#419626

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1586 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326822);