Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61006 |
| 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 | |
6660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
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)(21,25)(22,24)(23,27)(26,30)(28,32)(29,31)(33,34);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(21,28)(25,32)(29,34)(31,33);(5,12)(8,11)(13,18)(15,17)(20,24)(22,30)(25,31)(26,27)(29,34)(32,33);(4,9)(8,11)(10,16)(15,17)(19,21)(20,25)(22,29)(23,28)(24,31)(26,33)(27,32)(30,34);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(21,28)(25,32)(29,34)(31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483050);