Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#13603 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
878 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 878 conjugacy classes of subgroups — too many to draw. Only its 194 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)(5,13,16,29)(6,17,19,33)(7,21,23,11)(8,12)(9,15,25,31)(10,20)(14,28,22,30)(18,26,32,24)(27,34);(3,5)(4,9,12,24)(6,18)(7,22)(8,14)(10,13,27,28)(11,25,21,26)(15,19)(16,23)(17,29,33,30)(20,32)(31,34);(5,14)(9,24)(13,28)(15,18)(16,22)(25,26)(29,30)(31,32)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#152071, C2×#152093, C2×#512647, … |
| maximal normal subgroup of | C2×#152071, C2×#152093, C2×#512647, … |
| maximal quotient of | #72284551 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419829);