Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12117 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12117 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| 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 | |
1168 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1168 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)(5,16)(6,10)(7,14)(8,13)(9,15)(11,18)(12,20)(17,29)(19,30)(21,26)(22,28)(23,25)(24,33)(27,34)(31,32);(4,10)(6,20)(7,8)(11,24)(12,14)(13,25)(15,26)(16,29)(17,32)(18,22)(21,23)(27,30)(28,31)(33,34);(3,5)(4,11)(6,17)(7,22)(8,18)(9,19)(10,24)(12,29)(13,30)(14,16)(15,28)(20,32)(21,34)(23,33)(25,27)(26,31)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486345, C2×#485590, C2×#486077, … |
| maximal normal subgroup of | C2×#486345, C2×#485590, C2×#486077, … |
| maximal quotient of | #72265135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391425);