Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,12,20,26)(9,21)(10,14)(11,18)(13,22)(16,27,29,36)(17,28)(19,32,33,25)(23,34,35,30)(24,31);(1,2)(3,4)(5,7,9,18,10,17,22,31)(6,11,13,15,14,24,21,28)(8,16,19,30,20,29,33,34)(12,23,25,27,26,35,32,36);(2,4)(5,8,9,19,10,20,22,33)(6,12,13,25,14,26,21,32)(7,16,18,30,17,29,31,34)(11,23,15,27,24,35,28,36);(7,17)(11,24)(15,28)(16,29)(18,31)(23,35)(27,36)(30,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265255);