#504258

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=f,d^2=ei,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=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, 411454);