Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×#1419 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #357 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C23×D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
894 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 894 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2);(3,4);(5,6,8,14,11,16,10,27)(7,17,20,34,23,12,22,28)(9,24,19,31,26,32,25,15)(13,30,29,18,33,36,35,21);(6,12)(7,18)(8,10)(9,25)(13,31)(14,34)(15,33)(16,17)(19,26)(20,30)(21,23)(22,36)(24,29)(27,28)(32,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419895);