Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 170 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);(4,10,12,26)(5,15)(6,18)(7,22)(8,16)(11,19,23,27)(13,25,30,29)(14,31,24,28)(17,21)(20,32)(33,34);(3,4)(5,11)(6,19)(7,23)(8,24)(9,12)(10,17)(13,22)(14,16)(15,30)(18,26)(20,25)(21,27)(28,34)(29,33)(31,32);(3,5,16,22,9,7,8,15)(4,11,14,13,12,23,24,30)(6,18,33,32,21,17,20,34)(10,25,28,19,26,29,31,27)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216258, #72216315, #72216317, … |
| maximal normal subgroup of | #72216258, #72216315, #72216317, … |
| maximal quotient of | #72216311, #72216315, D3×#13658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419973);