#490863

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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

439 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 398059);