#491455

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

545 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,25,31)(19,32,34,33)(20,39,27,22)(21,40,41,28)(23,30,24,29)(26,42,43,35)(36,45,48,46)(37,44,38,47);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,22,36,23,37,21,35,25,34,39,48,24,38,41,42)(18,26,28,44,29,45,27,32,31,43,40,47,30,46,20,33);(2,7)(6,14)(8,12)(9,15)(17,20,21,29,24,40,22,18,25,27,41,30,23,28,39,31)(19,33,35,45,38,47,36,26,34,32,42,46,37,44,48,43);(19,34)(26,43)(32,33)(35,42)(36,48)(37,38)(44,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,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, 398651);