#147534

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

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×#135	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 54730);