#442197

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
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

637 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=f,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=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, 349393);