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×#60893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6142 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,23,28)(21,30,31,25)(22,33,34,27)(24,26,32,29);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,23,31)(20,25,28,30)(22,32,34,24)(26,27,29,33);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,26)(21,31)(24,32)(25,33)(27,30)(28,29);(4,10)(8,11)(9,18)(16,17)(19,22)(20,27)(21,32)(23,34)(24,31)(25,29)(26,30)(28,33);(21,31)(22,34)(25,30)(27,33)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482932);