Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#38045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7232 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);(3,4,5,6)(15,16)(17,25)(18,19)(20,22)(21,24)(23,30)(26,27)(28,29);(8,11)(9,13)(15,17,21,20)(16,22,24,25)(18,28,23,26)(19,27,30,29);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,18,21,23)(16,19,24,30)(17,26,20,28)(22,29,25,27);(15,19)(16,23)(17,27)(18,24)(20,29)(21,30)(22,26)(25,28);(7,9)(8,12)(10,13)(11,14)(15,20,21,17)(16,22,24,25)(18,28,23,26)(19,29,30,27)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542852);