Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#2017 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1912 conjugacy classes of subgroups; only the 326 normal subgroups are shown.
This group has 1912 conjugacy classes of subgroups — too many to draw. Only its 326 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,24)(17,34)(19,29)(20,28)(22,33)(23,32)(25,31)(27,35)(30,36);(5,7)(6,12)(8,22)(9,19)(10,18)(11,20)(13,30)(14,28)(15,29)(16,26)(17,23)(21,27)(24,35)(25,36)(31,34)(32,33);(5,8)(6,13)(7,17)(9,23)(10,24)(11,25)(12,27)(14,31)(15,32)(16,33)(18,22)(19,36)(20,35)(21,26)(28,30)(29,34)104 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2017⋊C3, C2×C4×#2017, #72217918 |
| maximal normal subgroup of | C4×#2017⋊C3, C2×C4×#2017, #72217918 |
| maximal quotient of | #72211651, C2×C4×#2017 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266187);