#508959

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eg,d^2=ei,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, 416155);