#409348

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

801 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)(3,8)(4,6)(5,7)(9,10,14,19,15,20,13,18)(11,21,22,16,25,29,24,31)(12,17,26,32,27,33,28,34)(23,37,36,40,38,35,39,30);(1,3,5,4)(2,6,7,8)(11,22,25,24)(16,29,31,21)(23,36,38,39)(30,37,40,35);(9,11)(10,16)(12,23)(13,22)(14,24)(15,25)(17,30)(18,29)(19,21)(20,31)(26,39)(27,38)(28,36)(32,35)(33,40)(34,37);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,23)(13,28)(14,26)(15,27)(16,30)(18,34)(19,32)(20,33)(21,35)(22,36)(24,39)(25,38)(29,37)(31,40)

Polycyclic presentation:

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

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, 316544);