#407936

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

986 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315132);