Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2684 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,15)(6,20)(7,13)(8,12)(9,14)(10,17)(11,22)(16,31)(18,27)(19,26)(21,34)(23,29)(24,30)(25,33)(28,32);(3,5,8,18,9,19,7,17)(4,10,13,26,14,27,12,15)(6,21,22,16,24,32,23,33)(11,28,20,31,29,34,30,25);(3,6)(4,11)(5,16)(7,22)(8,23)(9,24)(10,25)(12,20)(13,30)(14,29)(15,28)(17,32)(18,21)(19,33)(26,34)(27,31);(3,4)(5,15)(6,11)(7,13)(8,12)(9,14)(10,17)(16,28)(18,27)(19,26)(20,23)(21,31)(22,30)(24,29)(25,32)(33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411480);