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×#12312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3292 | C2 |
| 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 | |
1000 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1000 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);(3,4,8,12)(5,13,16,32)(6,18,22,34)(7,23,21,11)(9,24,26,15)(10,28,30,19)(14,29,17,31)(20,25,33,27);(6,19)(7,21)(10,18)(11,23)(13,29)(14,17)(15,20)(22,28)(24,33)(25,27)(30,34)(31,32);(3,5)(4,9)(6,20)(7,14)(8,16)(10,29)(11,25)(12,26)(13,18)(15,19)(17,21)(22,33)(23,27)(24,28)(30,31)(32,34)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#488869, #72226756, C2×#488870, … |
| maximal normal subgroup of | C2×#488869, #72226756, C2×#488870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396062);