#507300

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,40)(31,41)(33,45)(35,43)(37,47)(38,42)(39,46)(44,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,30,38,31,36,39,37,21,32,22,33,23,20,35,48)(18,24,40,45,41,34,46,44,26,29,27,42,28,25,43,47);(10,14)(13,15)(17,20,21,36)(18,25,26,34)(19,31,32,23)(22,38,30,33)(24,41,29,28)(27,45,40,42)(35,48,39,37)(43,47,46,44);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,26)(24,29)(25,34)(27,40)(28,41)(42,45)(43,46)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414496);