Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31280 |
| 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 | |
2258 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)(6,21)(7,13)(8,12)(9,14)(10,18)(11,23)(16,31)(17,20)(19,27)(22,32)(24,29)(25,30)(26,34)(28,33);(3,5,8,19,9,20,7,18)(4,10,13,17,14,27,12,15)(6,22,23,16,25,33,24,34)(11,28,21,31,29,32,30,26);(3,6)(4,11)(5,16)(7,23)(8,24)(9,25)(10,26)(12,21)(13,30)(14,29)(15,28)(17,32)(18,33)(19,22)(20,34)(27,31);(3,4)(5,17)(6,11)(7,13)(8,12)(9,14)(10,19)(15,20)(16,32)(18,27)(21,24)(22,26)(23,30)(25,29)(28,34)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411479);