Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1917 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)(7,18)(8,23,28,29)(9,14,32,19)(10,16)(11,17)(12,21)(13,27,20,26)(15,34)(22,35)(24,30,36,31)(25,33);(1,2)(3,4)(5,7,11,22)(6,12,17,25)(8,24,28,36)(9,29,32,23)(10,33,34,21)(13,30,20,31)(14,27,19,26)(15,35,16,18);(2,4)(5,8)(6,13)(7,19)(9,25)(10,26)(11,28)(12,32)(14,22)(15,23)(16,29)(17,20)(18,31)(21,36)(24,33)(27,34)(30,35);(5,9)(6,14)(7,20)(8,25)(10,31)(11,32)(12,28)(13,22)(15,36)(16,24)(17,19)(18,26)(21,23)(27,35)(29,33)(30,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290369);