Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20099 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,7)(10,17)(11,15)(12,16)(13,18)(19,20,23,26)(21,27,28,34)(22,30,32,25)(24,31,33,29);(3,4)(5,9)(6,13)(7,15)(8,11)(10,12)(14,17)(16,18)(21,28)(24,33)(27,34)(29,31);(3,5,8,7)(4,9,11,15)(6,12,14,16)(10,17,18,13)(19,21,23,28)(20,24,26,33)(22,31,32,29)(25,34,30,27);(3,6,8,14)(4,10,11,18)(5,12,7,16)(9,17,15,13)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266624);