#511739

order	512
nilpotency class	4
isoclinism	isoclinic class of #511738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32867
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,d^2=fh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418935);