Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#61111 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,14)(6,18)(7,11)(8,25)(9,12)(10,17)(13,26)(15,32)(16,20)(19,21)(22,31)(23,30)(24,29)(27,34)(28,33);(3,5)(4,10)(6,12)(7,21)(8,18)(9,16)(11,28)(13,20)(14,25)(15,22)(17,26)(19,31)(23,33)(24,34)(27,30)(29,32);(3,6,9,10)(4,5,12,16)(7,19,24,27)(8,20,26,14)(11,15,29,33)(13,17,25,18)(21,31,34,30)(22,32,23,28);(3,7)(4,11)(5,15)(6,19)(8,23)(9,24)(10,27)(12,29)(13,31)(14,32)(16,33)(17,34)(18,21)(20,28)(22,26)(25,30);(4,12)(5,16)(11,29)(13,25)(15,33)(17,18)(21,34)(30,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483372);