Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93822 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7083 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| prime radical | C23 | #359 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,12,7,15,27,44,9,16,24,45,8,33,28,56)(3,10,19,34,21,37,53,59,23,38,25,42,22,49,54,64)(5,29,30,57,31,14,48,43,6,32,26,55,18,13,46,47)(11,39,40,52,36,62,58,17,41,50,63,51,35,60,61,20);(2,10,16,38)(3,17)(4,24)(5,18)(6,31)(7,8)(11,14,41,13)(12,42,45,34)(15,49,33,37)(19,39)(20,23)(21,52)(22,51)(25,50)(26,46)(29,35,32,36)(30,48)(40,47,63,43)(44,59,56,64)(53,60)(54,62)(55,61,57,58)Polycyclic presentation:
2^9|a^2=d,b^2=gh,c^2=fhi,d^2=g,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1018);