Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12723 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2240 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)(5,14,17,29)(6,11,19,27)(7,12)(8,13)(9,15)(10,18,26,33)(16,22,20,34)(21,25)(23,30)(24,31)(28,32);(3,5,8,17)(4,9,13,24)(6,15,19,31)(7,20,21,16)(10,22,26,34)(11,14,27,29)(12,28,25,23)(18,32,33,30);(3,6,8,19)(4,10)(5,15,17,31)(7,18,21,33)(9,22)(11,12)(13,26)(14,28)(16,30,20,32)(23,29)(24,34)(25,27)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400643);