#410443

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 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)(3,6)(4,5)(7,8)(9,10,15,18)(11,19,23,32)(12,24,28,35)(13,14,29,30)(16,22,33,21)(17,26,36,27)(20,38,40,31)(25,34,39,37);(9,11,14,21,15,23,30,22)(10,16,13,19,18,33,29,32)(12,25,27,20,28,39,26,40)(17,34,24,31,36,37,35,38);(2,5)(6,8)(9,12,14,27,15,28,30,26)(10,17,13,24,18,36,29,35)(11,20,22,25,23,40,21,39)(16,31,32,34,33,38,19,37);(1,3,4,7)(2,6,5,8)(9,13,14,18,15,29,30,10)(11,19,21,33,23,32,22,16)(12,24,27,36,28,35,26,17)(20,37,39,38,40,34,25,31)

Polycyclic presentation:

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

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, 317639);