#492813

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=gi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

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, 400009);