#6432449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#400
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1833 conjugacy classes of subgroups; only the 824 normal subgroups are shown.

normal subgroups only

This group has 1833 conjugacy classes of subgroups — too many to draw. Only its 824 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 — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^b=hi,e^d=ei

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, 6339645);