#491988

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33892	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,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=de,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399184);