#401520

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,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, 308716);