#430635

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	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1161 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)(9,10,14,20,16,21,15,30)(11,19,23,33,24,32,13,17)(12,18,27,36,28,37,29,40)(22,35,38,34,25,39,26,31);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,25)(13,14)(15,23)(16,24)(18,34)(19,30)(20,32)(21,33)(22,28)(26,29)(27,38)(31,37)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(9,12,14,27,16,28,15,29)(10,18,20,36,21,37,30,40)(11,22,23,38,24,25,13,26)(17,31,19,35,33,34,32,39);(2,5)(6,8)(9,13,15,24,16,23,14,11)(10,19,30,17,21,32,20,33)(12,26,29,25,28,38,27,22)(18,35,40,31,37,39,36,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=efi,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, 337831);