#147963

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55159);