#491041

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	#33783	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

473 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:

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

Polycyclic presentation:

2^9|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, 398237);