#508467

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 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)(21,22,25,29,36,42,50,48,24,27,35,41,26,30,38,45)(23,31,33,46,39,52,51,40,32,47,49,43,34,44,37,28);(1,2)(3,4)(6,10)(11,17)(12,18)(19,20)(22,27)(28,40)(29,41)(30,42)(31,47)(43,46)(44,52)(45,48);(2,4)(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,23,24,32)(22,28,27,40)(25,37,35,51)(26,39,36,34)(29,44,41,52)(30,46,42,43)(31,48,47,45)(33,50,49,38);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(23,32)(28,40)(31,47)(33,49)(34,39)(37,51)(43,46)(44,52)

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^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415663);