#399291

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

979 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,15)(13,27)(14,19)(16,20)(17,23)(18,30)(22,36)(24,33)(25,32)(26,28)(29,34)(31,38)(35,39)(37,40);(2,5)(6,8)(9,11,15,24,16,25,14,23)(10,17,19,32,20,33,12,21)(13,22,29,38,30,39,28,37)(18,31,34,36,27,40,26,35);(9,12,14,20,16,19,15,10)(11,17,23,21,25,33,24,32)(13,26,28,27,30,34,29,18)(22,31,37,35,39,40,38,36);(1,3,4,7)(2,6,5,8)(9,13,14,28,16,30,15,29)(10,18,12,26,20,27,19,34)(11,22,23,37,25,39,24,38)(17,31,21,35,33,40,32,36)

Polycyclic presentation:

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

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, 306487);