Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C24 | C2×C16 |
| Frattini, cosocle | #1779 | C22 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1779 ◁ G | |
| LECS | G ▷ #1779 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,20,24)(19,25,26,32,27,22,28,30)(21,23,29,31);(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,30,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=i,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=de,f^b=fi,d^c=dg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 868);