#486290

order	512
nilpotency class	4
isoclinism	isoclinic class of #486155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

976 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,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, 393486);