Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2049 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1371 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);(5,6,11,16)(7,17,20,32)(8,21,25,30)(9,15,10,14)(12,26,28,19)(13,23,31,24)(18,29,22,27)(33,36,35,34);(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,29)(14,32)(15,17)(16,28)(18,25)(21,35)(23,34)(24,36)(27,31)(30,33);(5,8,9,23,11,25,10,24)(6,13,14,30,16,31,15,21)(7,18,19,34,20,22,26,36)(12,27,17,33,28,29,32,35)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266356);