Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
984 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 984 conjugacy classes of subgroups — too many to draw. Only its 154 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);(5,13)(6,17)(7,20)(9,23)(10,18)(11,22)(14,19)(15,31)(16,30)(21,27)(24,28)(25,34)(26,33)(29,32);(3,4,8,12)(5,9,15,25)(6,18,21,32)(7,22,20,11)(10,27,29,17)(13,28,31,33)(14,26,16,24)(19,23,30,34);(3,5)(4,9)(6,19)(7,14)(8,15)(10,28)(11,24)(12,25)(13,17)(16,20)(18,23)(21,30)(22,26)(27,31)(29,33)(32,34)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#488869, C2×#488872, #72224810, … |
| maximal normal subgroup of | C2×#488869, C2×#488872, #72224810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396063);