#506490

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413686);