#484163

order	512
nilpotency class	4
isoclinism	isoclinic class of #484162
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×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1279 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,32)(22,33)(23,29)(24,30)(25,38)(26,31)(27,35)(28,37)(34,40)(36,39);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,28)(12,20,23,34)(13,21,26,37)(16,27,29,39)(17,19,31,33)(24,35,25,36)(30,32,38,40);(9,12)(10,16)(11,20)(13,25)(14,23)(15,27)(17,30)(18,29)(19,32)(21,36)(22,34)(24,26)(28,39)(31,38)(33,40)(35,37);(12,23)(16,29)(20,34)(24,25)(27,39)(30,38)(32,40)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391359);