#491454

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19661	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

583 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398650);