Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2346 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);(3,4,9,15,10,16,8,14)(5,17,19,33,20,31,25,27)(6,21,23,12,7,13,22,30)(11,26,28,18,29,34,32,24);(3,5)(4,11)(6,18)(7,24)(8,25)(9,19)(10,20)(12,27)(13,31)(14,28)(15,32)(16,29)(17,21)(22,26)(23,34)(30,33);(3,6,10,7)(4,12,16,30)(5,18,20,24)(8,23,9,22)(11,27,29,33)(13,15,21,14)(17,28,31,32)(19,26,25,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400820);