Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29155 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2563 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1714 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);(3,4,8,14)(5,11)(6,18,19,24)(7,10,9,13)(12,21,28,20)(15,23)(16,30)(17,25)(22,27)(26,29)(31,34)(32,33);(4,10,14,13)(5,15,17,22)(6,19)(11,25)(12,18,28,24)(16,31,32,29)(20,21)(23,27);(3,5)(4,11)(6,16)(7,15)(8,17)(9,22)(10,23)(12,26)(13,27)(14,25)(18,30)(19,32)(20,29)(21,31)(24,33)(28,34);(3,6,7,20,8,19,9,21)(4,12,13,18,14,28,10,24)(5,16,15,29,17,32,22,31)(11,26,27,30,25,34,23,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375268);