#450920

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358116);