#6435891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6435890
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1141 conjugacy classes of subgroups; only the 488 normal subgroups are shown.

normal subgroups only

This group has 1141 conjugacy classes of subgroups — too many to draw. Only its 488 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343087);