#507309

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414505);