Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#2009 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1911 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);(5,6)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,23)(18,26)(20,31)(22,30)(24,29)(25,27)(28,34)(32,33)(35,36);(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,28)(14,31)(15,17)(16,27)(21,32)(22,35)(23,33)(24,34)(26,30)(29,36);(5,8,11,24)(6,13,16,30)(7,18,20,34)(9,23,10,22)(12,26,27,28)(14,21,15,29)(17,32,31,36)(19,35,25,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266266);