#491592

order	512
nilpotency class	4
isoclinism	isoclinic class of #491565
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

498 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,22,26)(19,27,28,42)(20,32,35,39)(21,36,33,25)(23,37,38,30)(24,29,41,34)(31,48,45,44)(40,47,43,46);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(19,28)(23,38)(27,42)(30,37)(31,45)(40,43)(44,48)(46,47);(17,19,29,46,21,31,20,30,22,28,34,47,33,45,35,37)(18,23,39,48,25,40,24,27,26,38,32,44,36,43,41,42);(3,10)(6,15)(9,16)(11,14)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398788);