Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 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)(5,6,11,16,12,18,10,17)(7,19,21,13,22,8,14,24)(9,25,28,34,29,33,27,15)(20,35,26,32,36,30,31,23);(5,7)(6,13)(8,16)(9,26)(10,14)(11,21)(12,22)(15,32)(17,19)(18,24)(20,27)(23,34)(25,30)(28,36)(29,31)(33,35);(2,4)(5,8,11,24,12,19,10,13)(6,14,16,7,18,21,17,22)(9,23,28,35,29,32,27,30)(15,31,25,20,34,26,33,36);(2,4)(5,9,10,27,12,29,11,28)(6,15,16,25,18,34,17,33)(7,20,21,26,22,36,14,31)(8,23,13,30,19,32,24,35)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295678);