#498028

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=deh,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405224);