#491038

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	#33780	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

458 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)(17,18,22,25)(19,23,28,38)(20,24,33,39)(21,34,32,27)(26,41,40,31)(29,35,42,47)(30,45,43,37)(36,48,46,44);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,38)(26,36)(27,37)(31,44)(32,43)(33,42)(34,45)(39,47)(40,46)(41,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(18,24)(20,31)(21,32)(23,35)(25,39)(26,27)(29,44)(30,43)(33,41)(34,40)(36,37)(38,47)(42,48)(45,46);(2,8)(3,9)(6,13)(10,15)(18,25)(19,28)(24,39)(26,40)(27,34)(29,42)(30,43)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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,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, 398234);