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×#3565 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4);(5,6,9,10)(7,13,14,11)(8,16,17,12)(15,18,20,19)(21,22)(23,28)(24,29)(25,33)(26,27)(30,34)(31,32)(35,36);(6,10)(7,14)(12,16)(15,20)(22,27)(24,28)(30,33)(32,35);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,35,33,32);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533384);