#398157

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1331 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)(9,10,13,17)(11,16,24,31)(12,25,26,34)(14,20,15,19)(18,28,32,27)(21,29,23,33)(22,36,38,39)(30,37,40,35);(2,5)(6,8)(9,11)(10,16)(12,22)(13,24)(14,21)(15,23)(17,31)(18,30)(19,33)(20,29)(25,36)(26,38)(27,35)(28,37)(32,40)(34,39);(10,17)(11,21,24,23)(16,29,31,33)(18,32)(19,20)(22,35,38,37)(25,34)(30,39,40,36);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,26)(14,27)(15,28)(16,30)(17,32)(19,25)(20,34)(21,35)(23,37)(24,38)(29,39)(31,40)(33,36)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=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, 305353);