#507139

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=i,d^2=fi,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, 414335);