#323445

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=egh,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

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, 230641);