#395818

order	512
nilpotency class	3
isoclinism	isoclinic class of #394127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1792 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)(9,10,15,20,14,19,16,32)(11,21,25,36,24,17,26,35)(12,27,29,37,13,30,31,18)(22,34,23,33,40,39,28,38);(2,5)(6,8)(9,11)(10,17)(12,28)(13,23)(14,24)(15,25)(16,26)(18,34)(19,21)(20,35)(22,29)(27,33)(30,38)(31,40)(32,36)(37,39);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,18,19,37)(11,22,24,40)(15,31,16,29)(17,33,21,38)(20,30,32,27)(23,25,28,26)(34,36,39,35);(9,13,14,12)(10,18,19,37)(11,23,24,28)(15,31,16,29)(17,34,21,39)(20,27,32,30)(22,25,40,26)(33,36,38,35)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 303014);