#496370

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1130 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)(3,7)(4,6)(5,8)(9,10,12,19,13,15,25,17)(11,20,24,34,21,36,22,37)(14,28,26,18,27,35,39,31)(16,29,32,23,30,40,33,38);(1,3,4,8)(2,5,6,7)(10,15)(11,21)(17,19)(18,31)(22,24)(23,38)(28,35)(29,40);(3,8)(5,7)(9,11)(10,16)(12,22)(13,21)(14,29)(15,30)(17,32)(18,34)(19,33)(20,35)(23,39)(24,25)(26,38)(27,40)(28,36)(31,37);(2,6)(5,7)(9,12,13,25)(10,17,15,19)(11,22,21,24)(14,26,27,39)(16,32,30,33)(18,28,31,35)(20,34,36,37)(23,29,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403566);