#474590

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=fi,g^d=gi,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, 381786);