#491851

order	512
nilpotency class	4
isoclinism	isoclinic class of #491712
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	#33872	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399047);