Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15,19,29)(8,20)(9,13)(10,14)(11,18,27,24)(12,22)(16,25)(17,30,21,35)(23,34)(26,32,33,36)(28,31);(1,2)(3,4)(5,7,16,18,10,19,9,24)(6,11,25,15,14,27,13,29)(8,21,31,36,22,17,23,32)(12,26,34,30,20,33,28,35);(1,3,4,2)(5,8,9,23,10,22,16,31)(6,12,13,28,14,20,25,34)(7,17,18,32,19,21,24,36)(11,26,15,30,27,33,29,35);(8,22)(12,20)(17,21)(23,31)(26,33)(28,34)(30,35)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265565);