Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19039 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3568 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,11)(5,12,15,18)(6,10,7,16)(9,13,17,14)(19,20)(21,28)(22,26)(23,31)(24,27)(25,29)(30,33)(32,34);(3,5)(4,9)(6,14)(7,13)(8,15)(10,18)(11,17)(12,16)(19,20)(21,25)(22,26)(23,31)(24,27)(28,32)(29,33)(30,34);(3,4)(5,9)(6,10)(7,16)(8,11)(12,13)(14,18)(15,17)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258628);