#147960

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	C₂×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 55156);