#429601

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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)(3,7,8,5)(9,10,14,20,15,21,16,22)(11,19,13,30,24,33,25,17)(12,26,28,36,29,18,31,35)(23,37,27,32,38,34,39,40);(1,3,4,8)(2,5,6,7)(9,11,15,24)(10,17,21,30)(12,27,29,39)(13,14,25,16)(18,34,26,37)(19,20,33,22)(23,31,38,28)(32,36,40,35);(2,6)(5,7)(9,12)(10,18)(11,23)(13,27)(14,28)(15,29)(16,31)(17,32)(19,34)(20,35)(21,26)(22,36)(24,38)(25,39)(30,40)(33,37);(3,8)(5,7)(9,13,14,24,15,25,16,11)(10,19,20,30,21,33,22,17)(12,27,28,38,29,39,31,23)(18,34,35,40,26,37,36,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=efi,e^2=i,g^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^b=gi,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, 336797);