#508470

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

481 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)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,22,26,31,39,49,52,42,25,30,38,48,27,32,41,45)(23,33,34,47,37,44,40,29,24,28,36,43,35,50,51,46);(1,2)(3,4)(6,10)(11,17)(12,18)(19,20)(21,23)(22,28)(24,25)(26,34)(27,35)(29,45)(30,33)(31,43)(32,44)(36,38)(37,39)(40,52)(41,51)(42,46)(47,48)(49,50);(2,4)(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,25,23)(22,29,30,46)(26,40,38,51)(27,35,39,37)(28,42,33,45)(31,44,48,50)(32,43,49,47)(34,41,36,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(23,24)(28,33)(29,46)(34,36)(35,37)(40,51)(43,47)(44,50)

Polycyclic presentation:

2^9|a^2=f,c^2=e,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, 415666);