#135676

order	512
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#486
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8787
prime radical	C₂³	#430
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

195 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,11,7,13,31,43,8,14,32,44,30,12,5,29)(3,15,19,35,20,47,17,9,21,34,52,36,51,46,18,37)(4,22,26,40,27,56,25,10,28,39,61,41,60,55,24,42)(16,38,49,64,23,57,48,45,50,63,58,54,59,33,53,62);(1,3,5,18,30,51,32,52,8,21,31,17,7,20,6,19)(2,9,11,34,13,36,43,46,14,37,44,15,12,35,29,47)(4,23,25,58,60,16,26,48,28,59,24,49,27,50,61,53)(10,38,22,54,41,57,56,62,42,63,39,64,40,33,55,45);(1,4,5,24,30,60,32,61,8,28,31,25,7,27,6,26)(2,10,11,39,13,41,43,55,14,42,44,22,12,40,29,56)(3,16,17,48,51,59,19,49,21,50,18,53,20,23,52,58)(9,33,15,45,36,38,47,54,37,57,34,62,35,63,46,64)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42872);