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×#62367 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6170 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)(21,28)(22,26)(23,32)(24,27)(25,31)(29,34)(30,33);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(21,29)(25,33)(28,34)(30,31);(19,21,23,31,24,29,22,30)(20,25,26,28,27,33,32,34);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22,24,23)(20,26,27,32)(21,30,29,31)(25,28,33,34);(4,10)(8,11)(9,18)(16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485593);