Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12170 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12170 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
936 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 936 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14)(6,9)(7,22)(8,13)(10,16)(11,18)(12,20)(15,28)(17,27)(19,23)(21,24)(25,34)(26,33)(29,31)(30,32);(4,9)(6,18)(7,20)(10,23)(11,12)(13,24)(14,28)(15,31)(17,32)(19,33)(21,34)(22,25)(26,27)(29,30);(3,5,8,16)(4,10)(6,19,21,15)(7,17,20,32)(9,23)(11,26)(12,27)(13,14)(18,33,34,31)(22,30)(24,28)(25,29)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486335, C2×#485590, C2×#486084, … |
| maximal normal subgroup of | C2×#486335, C2×#485590, C2×#486084, … |
| maximal quotient of | #72265113, #72265117 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391844);