#496948

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3035	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404144);