#492551

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
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×D₈ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgi,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399747);