#415411

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322607);