#423386

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1301 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efi,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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, 330582);