#491034

order	512
nilpotency class	4
isoclinism	isoclinic class of #491025
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	#33778	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

446 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,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,d^c=dg,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, 398230);