#489594

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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	#62325	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18,22,26)(19,28,31,47)(20,33,21,38)(23,41,36,27)(24,29,43,30)(25,37,35,40)(32,48,42,45)(34,44,39,46);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19,22,31)(18,24,26,43)(20,34,21,39)(23,42,36,32)(25,45,35,48)(27,46,41,44)(28,33,47,38)(29,40,30,37);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,18)(19,24)(20,35)(21,25)(22,26)(23,27)(28,30)(29,47)(31,43)(32,44)(33,37)(34,48)(36,41)(38,40)(39,45)(42,46);(2,9)(3,12)(8,10)(13,14)(17,20,36,40,22,21,23,37)(18,25,41,33,26,35,27,38)(19,29,42,39,31,30,32,34)(24,28,46,48,43,47,44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396790);