#129404

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8303
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

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, 36600);