#415688

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 322884);