#476068

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
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

694 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)(11,17)(12,25,28,18)(13,29)(14,21)(15,31)(16,20)(19,24)(22,35,38,32)(23,33)(26,40,39,34)(27,37,30,36);(2,5)(6,8)(9,11,14,23,16,24,15,13)(10,17,21,33,20,19,31,29)(12,26,27,38,28,39,30,22)(18,34,36,35,25,40,37,32);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,26)(14,30)(15,27)(16,28)(17,32)(19,35)(20,25)(21,37)(23,39)(24,38)(29,34)(31,36)(33,40);(3,7)(6,8)(9,13,15,24,16,23,14,11)(10,19,21,29,20,17,31,33)(12,26,27,38,28,39,30,22)(18,35,37,34,25,32,36,40)

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^c=de,g^c=gi

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, 383264);