#404394

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi,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, 311590);