#490292

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
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

663 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)(19,28)(20,33)(21,38)(22,41)(23,27)(24,30)(25,40)(26,36)(29,44)(31,48)(32,47)(34,43)(35,37)(39,46)(42,45);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,11)(10,16)(17,19,23,32)(18,24,27,44)(20,34,21,39)(22,42,36,31)(25,45,35,48)(26,46,41,43)(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);(3,12)(7,16)(11,15)(13,14)(17,20,36,40,23,21,22,37)(18,25,41,33,27,35,26,38)(19,29,42,39,32,30,31,34)(24,28,46,48,44,47,43,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dghi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397488);