#491081

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	#33813	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

436 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)(3,11,13,14)(4,8,7,16)(5,12,15,9)(17,18,24,30)(19,29,33,41)(20,34,36,40)(21,28,22,27)(23,38,32,25)(26,42,44,31)(35,46,37,48)(39,43,47,45);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14)(17,19,22,32,24,33,21,23)(18,25,28,29,30,38,27,41)(20,31,35,47,36,42,37,39)(26,40,43,46,44,34,45,48);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,39)(24,36)(25,40)(27,45)(28,43)(29,46)(30,44)(32,47)(33,42)(34,38)(41,48);(3,13)(4,7)(8,16)(11,14)(17,21,24,22)(18,27,30,28)(19,23,33,32)(20,35,36,37)(25,41,38,29)(26,43,44,45)(31,47,42,39)(34,48,40,46)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^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,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, 398277);