#508926

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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

684 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 162 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)(7,10)(8,9)(11,12)(13,14,18,23,36,48,64,60,19,24,38,50,17,22,35,47)(15,20,27,41,26,40,55,68,29,43,58,71,28,42,57,70)(16,30,32,44,31,61,63,46,34,62,65,45,33,49,37,21)(25,51,53,72,59,76,75,67,54,73,74,66,52,69,56,39);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,28)(18,27)(19,29)(21,39)(22,42)(23,41)(24,43)(26,36)(30,51)(31,59)(32,53)(33,52)(34,54)(35,57)(37,56)(38,58)(40,48)(44,72)(45,66)(46,67)(47,70)(49,69)(50,71)(55,64)(60,68)(61,76)(62,73)(63,75)(65,74);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,19,34)(14,21,24,46)(15,25,29,54)(17,31,36,33)(18,37,38,63)(20,39,43,67)(22,44,48,45)(23,49,50,61)(26,52,28,59)(27,56,58,75)(30,60,62,47)(32,64,65,35)(40,66,42,72)(41,69,71,76)(51,68,73,70)(53,55,74,57);(7,11)(10,12)(13,17,19,36)(14,22,24,48)(15,26,29,28)(16,31,34,33)(18,35,38,64)(20,40,43,42)(21,44,46,45)(23,47,50,60)(25,52,54,59)(27,55,58,57)(30,61,62,49)(32,63,65,37)(39,66,67,72)(41,68,71,70)(51,69,73,76)(53,56,74,75)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,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, 416122);