#508957

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, 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

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=i,d^2=e,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, 416153);