#508473

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

485 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)(21,22,24,30,36,44,48,42,26,29,37,43,25,31,35,28)(23,27,33,39,46,51,52,50,32,41,45,49,34,40,47,38);(1,2)(3,4)(6,10)(11,17)(12,18)(19,20)(21,23)(22,27)(24,33)(25,34)(26,32)(28,38)(29,41)(30,39)(31,40)(35,47)(36,46)(37,45)(42,50)(43,49)(44,51)(48,52);(2,4)(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,28)(24,35)(25,36)(27,38)(29,42)(30,31)(33,47)(34,46)(37,48)(39,40)(41,50)(43,44)(45,52)(49,51);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,29)(23,32)(28,42)(30,43)(31,44)(33,45)(34,46)(47,52)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 415669);