Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C4×#3552 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3552 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4);(5,6)(7,8)(9,10)(11,12)(14,19)(15,16)(17,25)(27,28);(6,10)(7,11)(13,14,18,19)(15,21,16,20)(17,22,25,26)(23,28,24,27);(13,15,18,16)(14,20,19,21)(17,23,25,24)(22,27,26,28);(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,21,19,20)(17,24,25,23)(22,28,26,27);(5,8)(6,11)(7,10)(9,12)(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533386);