#510847

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

517 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)(18,24)(19,29)(26,28)(27,40)(31,38)(32,41)(35,43)(36,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,24)(19,30,29,46)(20,28,23,26)(22,35,34,43)(25,41,39,32)(27,33,40,37)(31,47,38,44)(36,45,48,42);(1,4,5,6)(2,3,7,8)(10,14)(13,15)(17,19,21,29)(18,25,24,39)(20,31,23,38)(22,36,34,48)(26,42,28,45)(27,44,40,47)(30,43,46,35)(32,33,41,37);(2,7)(3,8)(11,16)(13,15)(17,20,21,23)(18,26,24,28)(19,31,29,38)(22,33,34,37)(25,42,39,45)(27,43,40,35)(30,44,46,47)(32,36,41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=efi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 418043);