Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1926 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,16,13,17,12,30)(7,18,23,34,21,36,29,32)(8,9,24,10,25,27,15,26)(14,28,19,20,31,22,33,35);(1,2,3,4)(5,7,9,20,12,23,26,22,13,21,27,35,11,29,10,28)(6,14,8,18,16,33,24,32,17,31,25,36,30,19,15,34);(5,8)(6,9)(7,19)(10,16)(11,24)(12,15)(13,25)(14,29)(17,27)(18,35)(20,36)(21,33)(22,32)(23,31)(26,30)(28,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55586);