Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13578 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13580 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1110 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1110 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)(5,15,18,25)(6,11)(7,22)(8,13,24,29)(9,14)(10,17,26,34)(12,23,30,21)(16,31,33,27)(19,28,20,32);(3,5,9,20)(4,10,14,27)(6,17,7,16)(8,18,24,19)(11,28,22,25)(12,15,30,32)(13,31,29,26)(21,34,23,33);(3,6,9,7)(4,11,14,22)(5,16,20,17)(8,21,24,23)(10,25,27,28)(12,29,30,13)(15,31,32,26)(18,33,19,34)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#152119, C2×#153150, C2×#153148, … |
| maximal normal subgroup of | C2×#152119, C2×#153150, C2×#153148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419789);