#140141

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9545
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

237 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47337);