#512463

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 76 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,21,28)(15,29,36,63)(16,37,40,24)(17,23,44,58)(18,45,47,26)(19,48,30,25)(20,49,46,27)(22,50,55,33)(31,60,67,74)(32,59,41,52)(34,54,64,61)(35,53,68,62)(38,57,43,69)(39,56,42,70)(51,66,75,71)(65,76,72,73);(6,9)(10,12)(13,15,30,64,20,35,47,50,21,36,19,34,46,68,18,33)(14,22,45,62,27,54,48,63,28,55,26,53,49,61,25,29)(16,31,43,72,39,66,44,32,40,67,38,65,42,71,17,41)(23,51,70,76,57,74,24,59,58,75,56,73,69,60,37,52);(1,2,3,4)(13,16,30,43,20,39,47,44,21,40,19,38,46,42,18,17)(14,23,45,70,27,57,48,24,28,58,26,56,49,69,25,37)(15,31,64,72,35,66,50,32,36,67,34,65,68,71,33,41)(22,51,62,76,54,74,63,59,55,75,53,73,61,60,29,52);(5,7,8,11)(6,10,9,12)(13,17,18,42,46,38,19,40,21,44,47,39,20,43,30,16)(14,24,25,57,49,70,26,23,28,37,48,69,27,56,45,58)(15,32,33,66,68,72,34,31,36,41,50,71,35,65,64,67)(22,52,29,60,61,73,53,75,55,59,63,74,54,76,62,51)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fh,d^2=e,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,d^b=di

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, 419659);