Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7535 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
| direct product | C2×#7536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1409 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 50 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)(5,14,16,30)(6,18,20,10)(7,12,23,11)(8,13)(9,15,24,32)(17,31,22,27)(19,25)(21,28,34,29)(26,33);(3,5)(4,9,13,24)(6,19)(7,22)(8,17)(10,27,29,30)(11,26,12,25)(14,28,31,18)(15,20)(16,23)(21,33)(32,34)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153547, C2×#153641, C2×#153639, … |
| maximal normal subgroup of | C2×#153547, C2×#153641, C2×#153639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60728);