#401521

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

753 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:

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

Polycyclic presentation:

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

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, 308717);