#398156

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1291 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)(9,10,15,20)(11,16,23,31)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,32,24,29)(22,36,37,40)(30,38,39,35);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,24)(15,23)(17,30)(18,29)(19,32)(20,31)(25,36)(26,35)(27,37)(28,38)(33,39)(34,40);(11,21,23,24)(16,29,31,32)(22,35,37,38)(30,36,39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,30)(18,25)(19,34)(20,33)(21,35)(23,37)(24,38)(29,36)(31,39)(32,40)

Polycyclic presentation:

2^9|a^2=i,c^2=f,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, 305352);