#490851

order	512
nilpotency class	4
isoclinism	isoclinic class of #490801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dgi,f^c=fh,h^c=hi

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, 398047);