Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94193 |
| 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 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,27,40,29,42,7,13,30,43,8,14,33,46)(3,15,17,47,52,59,25,57,19,38,54,61,20,48,56,64)(4,12,22,44,6,31,32,60,23,10,26,39,24,45,28,58)(9,34,35,16,49,50,41,55,36,53,62,21,37,51,63,18);(1,3,6,18,27,52,23,16,7,19,24,55,8,20,4,21)(2,9,12,15,14,37,45,48,13,36,10,38,40,49,31,59)(5,25,28,51,29,54,22,34,30,56,32,50,33,17,26,53)(11,41,39,64,46,35,60,61,43,63,44,57,42,62,58,47)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=efg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1464);