#458326

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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)(4,8)(5,7)(9,10)(11,23)(12,29)(13,18)(14,21)(15,20)(16,22)(17,26)(19,30)(24,35)(25,37)(27,34)(28,33)(31,36)(32,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11,15,27,16,28,14,26)(10,17,21,33,22,34,20,23)(12,24,13,25,31,40,30,39)(18,35,29,38,19,32,36,37);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,24)(13,14)(15,30)(16,31)(17,32)(19,22)(20,29)(21,36)(23,37)(25,26)(27,39)(28,40)(33,38)(34,35);(9,13,15,31,16,30,14,12)(10,19,21,36,22,18,20,29)(11,25,27,40,28,39,26,24)(17,32,33,37,34,35,23,38)

Polycyclic presentation:

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

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, 365522);