#495663

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

457 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,16,11,14,21,34,24,17,22,33,25,15,20,13,19)(12,26,31,27,29,18,28,37,32,38,35,36,30,39,40,23);(9,11,16,21,14,24,34,22,17,25,33,20,15,19,13,10)(12,27,31,18,29,37,28,38,32,36,35,39,30,23,40,26);(2,5)(6,8)(9,12,14,29,17,32,15,30)(10,18,20,26,22,39,21,38)(11,23,19,36,25,37,24,27)(13,28,16,35,34,40,33,31);(1,3,4,7)(2,6,5,8)(9,13,15,33,17,34,14,16)(10,19,20,25,22,24,21,11)(12,28,30,31,32,40,29,35)(18,36,26,37,39,27,38,23)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=ch,e^b=ei,g^c=gi

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, 402859);