Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394590 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17)(8,19,24,13)(9,25)(10,15,29,34)(12,28)(14,27)(18,26)(20,31)(21,33)(22,36)(23,35,30,32);(1,2)(3,4)(5,7,11,21)(6,12,16,31)(8,22,24,18)(9,19,27,13)(10,28,29,20)(14,30,25,23)(15,33,34,17)(26,32,36,35);(2,4)(5,8,10,23,11,24,29,30)(6,13,15,32,16,19,34,35)(7,18,20,14,21,22,28,25)(9,17,26,31,27,33,36,12);(5,9)(6,14)(7,19)(8,17)(10,26)(11,27)(12,30)(13,21)(15,22)(16,25)(18,34)(20,35)(23,31)(24,33)(28,32)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302982);