Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94192 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1391 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | #2960 | C22 |
| socle | C2 | #7248 |
| prime radical | C23 | #338 |
| LCS | G ▷ C22×C8 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2960 ◁ G | |
| LECS | G ▷ #2960 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,26,39,28,41,7,12,29,42,8,13,32,43)(3,14,16,44,52,64,24,57,18,45,53,60,19,46,55,63)(4,20,21,40,6,30,31,59,22,10,25,37,23,56,27,58)(9,33,34,15,47,48,38,17,35,54,61,49,36,51,62,50);(1,3,6,17,26,52,22,49,7,18,23,50,8,19,4,15)(2,9,10,14,13,36,30,46,12,35,20,45,39,47,56,64)(5,24,27,48,28,53,21,54,29,55,31,51,32,16,25,33)(11,38,40,63,43,34,58,60,42,62,37,57,41,61,59,44)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=fg,e^2=fghi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1462);