Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62546 |
| 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 | |
6684 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)(5,12)(6,10)(7,16)(8,11)(9,14)(13,18)(15,17)(19,20,23,27)(21,28,29,34)(22,25,30,26)(24,32,33,31);(3,5)(4,9)(6,13)(7,15)(8,14)(10,17)(11,12)(16,18)(21,29)(24,33)(28,34)(31,32);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,12,18)(19,21,30,32,23,29,22,31)(20,24,25,34,27,33,26,28);(5,14)(9,12)(13,15)(17,18)(19,22,23,30)(20,25,27,26)(21,31,29,32)(24,34,33,28);(3,7,8,6)(4,10,11,16)(5,15,14,13)(9,17,12,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10487156);