#137833

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂³	#435
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45029);