#504666

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	#13319	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

535 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,15)(13,28)(14,20)(16,19)(17,21)(18,25)(23,31)(24,38)(26,35)(27,34)(29,32)(30,36)(33,37)(39,40);(1,3,5,4)(2,6,7,8)(9,11,15,26,17,27,14,25)(10,18,20,34,21,35,12,22)(13,24,29,40,30,37,16,31)(19,33,36,39,32,38,28,23);(1,4,5,3)(2,6,7,8)(9,12,14,21,17,20,15,10)(11,23,25,38,27,39,26,33)(13,28,16,32,30,36,29,19)(18,31,22,37,35,40,34,24);(2,7)(6,8)(9,13,15,29,17,30,14,16)(10,19,20,36,21,32,12,28)(11,24,26,40,27,37,25,31)(18,33,34,39,35,38,22,23)

Polycyclic presentation:

2^9|b^2=ei,c^2=e,d^2=f,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, 411862);