Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1824 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,18,11,17,27,19)(7,20,8,24,22,36,23,35)(9,26,28,15,10,29,32,16)(13,25,14,31,33,30,34,21);(1,2)(3,4)(5,7,12,8,11,22,27,23)(6,13,19,34,17,33,18,14)(9,25,28,31,10,30,32,21)(15,35,26,36,16,24,29,20);(2,4)(5,8,12,22,11,23,27,7)(6,14,19,13,17,34,18,33)(9,21,28,25,10,31,32,30)(15,36,26,24,16,20,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56519);