#430636

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1121 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:

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

Polycyclic presentation:

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

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, 337832);