Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60993 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20,23,26)(21,28,31,34)(22,27,24,32)(25,33,29,30);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,20)(21,29)(22,32)(23,26)(24,27)(25,31)(28,33)(30,34);(5,12)(8,11)(13,18)(15,17)(19,21,23,31)(20,25,26,29)(22,30,24,33)(27,34,32,28);(4,9)(8,11)(10,16)(15,17)(20,26)(25,29)(27,32)(28,34);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(19,22,23,24)(20,27,26,32)(21,30,31,33)(25,34,29,28)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483032);