Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33794 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1416 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)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(20,24)(21,27)(23,31)(25,33)(26,30)(28,32)(29,34);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,20,22,24)(21,28,29,25)(23,26,30,31)(27,33,34,32);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21,22,29)(20,25,24,28)(23,32,30,33)(26,34,31,27);(4,10)(8,11)(9,18)(16,17)(19,20,22,24)(21,25,29,28)(23,26,30,31)(27,32,34,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530594);