Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2193 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#447 |
| prime radical | C2×#2193 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(6,12)(7,16)(8,19)(9,10)(13,27)(14,31)(15,26)(17,33)(18,23)(20,24)(21,22)(25,28)(29,36)(30,34);(5,6)(7,13)(8,20)(9,14)(10,24)(11,15)(12,19)(16,28)(17,34)(18,30)(21,26)(22,31)(23,29)(25,32)(27,33)(35,36);(5,7,11,18)(6,13,15,30)(8,17,22,33)(9,23,10,16)(12,25,26,36)(14,29,24,28)(19,32,21,35)(20,34,31,27)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402909);