#494856

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402052);