#476069

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

684 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)(11,21)(12,17,25,36)(13,28)(14,18)(15,19)(16,23)(20,31)(22,35,39,32)(24,40,29,34)(26,37,27,38)(30,33);(2,5)(6,8)(9,11,15,13,14,23,31,30)(10,16,19,33,18,21,20,28)(12,24,26,39,25,29,27,22)(17,34,37,35,36,40,38,32);(1,3,4,7)(2,6,5,8)(9,12,14,25)(10,17,18,36)(11,22,23,39)(13,29,30,24)(15,27,31,26)(16,32,21,35)(19,38,20,37)(28,34,33,40);(3,7)(6,8)(9,13,15,23,14,30,31,11)(10,16,20,28,18,21,19,33)(12,24,27,22,25,29,26,39)(17,35,37,40,36,32,38,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383265);