#6369751

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=i,h^2=i,b^a=bf,f^a=fi,h^a=hi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276947);