Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8,12,14)(9,13,10,11)(15,16)(17,21)(18,26)(19,20)(22,27)(23,25)(24,29)(28,30);(7,9)(8,13)(10,12)(11,14)(16,20)(18,27)(21,23)(24,30);(3,4)(5,6)(8,14)(9,10)(15,16)(17,23)(18,22)(19,20)(21,25)(24,29)(26,27)(28,30);(7,10,12,9)(8,11,14,13)(15,17,19,25)(16,21,20,23)(18,24,27,30)(22,28,26,29);(7,11)(8,9)(10,14)(12,13)(15,18,19,27)(16,22,20,26)(17,24,25,30)(21,28,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613860);