Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5330 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);(4,8)(6,9)(11,12)(13,17)(14,16)(15,18)(19,20,24,25)(21,22,26,29)(23,31,32,27)(28,34,33,30);(3,4)(5,9)(6,10)(7,8)(11,13,16,15)(12,17,14,18)(20,25)(22,29)(27,31)(30,34);(4,8)(5,10)(11,14,16,12)(13,17,15,18)(19,21,24,26)(20,22,25,29)(23,28,32,33)(27,30,31,34);(3,5)(4,6)(7,10)(8,9)(11,15,16,13)(12,17,14,18)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34);(3,6)(4,5)(7,9)(8,10)(19,23)(20,27)(21,28)(22,30)(24,32)(25,31)(26,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006934);