#135656

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#335
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

252 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42852);