#507138

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=i,d^2=f,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, 414334);