Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12864 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C22 | #1820 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14,17,12)(6,18)(7,11)(8,13)(9,16,23,30)(10,15,25,29)(19,22)(20,27)(21,26)(24,28,33,34)(31,32);(3,5)(4,9,13,23)(6,15)(7,16)(8,17)(10,22,25,32)(11,14,26,12)(18,24,27,33)(19,28)(20,29)(21,30)(31,34);(3,6,7,19,8,20,21,31)(4,10,11,24,13,25,26,33)(5,15,16,28,17,29,30,34)(9,22,14,27,23,32,12,18)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401087);