#489468

order	512
nilpotency class	4
isoclinism	isoclinic class of #489365
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396664);