#486510

order	512
nilpotency class	4
isoclinism	isoclinic class of #486147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#18380	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

870 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dgi,e^b=eg,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393706);