Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1688 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);(3,4)(5,12)(6,10)(7,16)(8,11)(9,13)(14,17)(15,18)(19,20,24,29)(21,30,22,26)(23,28,25,33)(27,31,34,32);(3,5,7,13,8,14,6,15)(4,9,10,12,11,18,16,17)(19,21,24,22)(20,26,29,30)(23,32,25,31)(27,28,34,33);(19,22,24,21)(20,27,29,34)(23,32,25,31)(26,28,30,33);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,18,12)(19,23,24,25)(20,28,29,33)(21,31,22,32)(26,27,30,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261768);