Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7115 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #376 | C23 |
| socle | C22 | #1270 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 602 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 — 3 generators on 34 points:
(1,2);(3,4)(5,13,15,28,16,25,20,34)(6,17,18,32)(7,11)(8,12)(9,14,22,29,24,31,23,30)(10,19,26,33)(21,27);(3,5)(4,9)(6,14)(7,20)(8,16)(10,25)(11,22)(12,24)(13,26)(15,21)(17,28)(18,31)(19,29)(23,27)(30,33)(32,34)80 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#7115, #72213050, #72213162, … |
| maximal normal subgroup of | C22×#7115, #72213050, #72213162, … |
| maximal quotient of | C2×#93376, #72211373, C2×#92837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31255);