#490305

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
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	#62341	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

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

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, 397501);