Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#13607 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
870 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 870 conjugacy classes of subgroups — too many to draw. Only its 186 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)(5,13,15,28)(6,17,18,33)(7,20,22,11)(8,12)(9,14,24,31)(10,19)(16,26,21,29)(23,30,25,32)(27,34);(3,5,8,16)(4,9,12,23)(6,14,18,30)(7,21,22,15)(10,26,27,13)(11,24,20,25)(17,29,33,28)(19,31,34,32);(6,18)(9,23)(10,27)(14,30)(17,33)(19,34)(24,25)(31,32)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512647, C2×#152134, C2×#152136, … |
| maximal normal subgroup of | C2×#512647, C2×#152134, C2×#152136, … |
| maximal quotient of | #72284690 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419839);