Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,12,20,28,18,26,19,27)(8,13,23,31,22,30,24,32)(17,33,21,36,34,25,35,29);(2,4)(5,7)(6,12)(8,21,22,35)(9,19)(10,20)(11,18)(13,29,30,36)(14,27)(15,28)(16,26)(17,23,34,24)(25,31,33,32);(5,8)(6,13)(7,17)(9,24)(10,23)(11,22)(12,25)(14,32)(15,31)(16,30)(18,34)(19,35)(20,21)(26,33)(27,36)(28,29);(2,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,25)(14,27)(15,28)(16,26)(21,23)(22,34)(24,35)(29,31)(30,33)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273515);