Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C4×#106 |
| prime radical | C24 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,19,22,31)(8,15)(9,27)(10,28)(11,17)(12,29)(13,18)(14,21,30,23)(16,26)(20,32,35,33)(24,34,25,36);(2,4)(5,7,10,23,12,25,26,35,13,22,9,21,11,24,8,20)(6,14,16,31,17,32,28,36,18,30,15,19,29,33,27,34);(5,8,11,9,13,26,12,10)(6,15,17,27,18,16,29,28)(7,20,24,21,22,35,25,23)(14,19,32,34,30,31,33,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dh,f^a=fh,h^a=hi,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53531);