#489724

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61091	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396920);