#491622

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	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

552 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)(6,11)(8,13)(10,15)(14,16)(17,18)(19,27)(20,23)(21,34)(22,36)(24,30)(25,35)(26,33)(28,41)(29,38)(31,48)(32,39)(37,44)(40,46)(42,47)(43,45);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(18,23)(19,28)(25,38)(26,36)(30,44)(31,45)(34,39)(46,47);(2,7)(6,14)(8,12)(9,15)(17,19,29,46,22,31,21,30,20,28,35,47,33,45,32,44)(18,24,39,48,26,40,25,27,23,37,34,43,36,42,38,41);(3,10)(6,14)(9,15)(11,16)

Polycyclic presentation:

2^9|b^2=ei,c^2=fghi,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, 398818);