#510869

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,27,29,24)(20,25,32,38)(22,26,34,37)(28,43,45,39)(30,44,41,35)(31,46,47,40)(33,36,42,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,24,23,27)(20,31,32,47)(22,35,34,44)(25,40,38,46)(26,41,37,30)(28,42,45,33)(36,39,48,43);(1,4,5,6)(2,3,7,8)(10,14)(13,15)(17,20,21,32)(18,25,23,38)(19,28,29,45)(22,36,34,48)(24,39,27,43)(26,42,37,33)(30,46,41,40)(31,44,47,35);(2,7)(3,8)(11,16)(13,15)(17,18,21,23)(19,27,29,24)(20,25,32,38)(22,26,34,37)(28,43,45,39)(30,44,41,35)(31,46,47,40)(33,36,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418065);