#498780

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

441 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 214 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefi,d^b=dei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405976);