Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33854 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,10)(5,15)(6,18)(7,8)(11,28)(12,30)(13,32)(14,27)(16,33)(17,23)(19,24)(21,22)(25,29)(26,31);(3,4)(5,11)(6,19)(7,13)(8,24)(9,14)(10,18)(12,16)(15,29)(17,30)(20,25)(21,27)(22,32)(23,26)(28,33)(31,34);(3,5)(4,11)(6,16)(7,23)(8,15)(9,17)(10,25)(12,19)(13,26)(14,30)(18,20)(21,34)(22,33)(24,29)(27,31)(28,32);(3,5)(4,12)(6,20)(7,15)(8,23)(9,17)(10,26)(11,19)(13,25)(14,28)(16,18)(21,33)(22,34)(24,31)(27,29)(30,32)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530849);