#506286

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

647 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

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, 413482);