#140113

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9541
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

247 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47309);