Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,11)(7,16)(8,12)(13,27)(14,23)(15,24)(17,32)(18,31)(19,26)(20,33)(22,28)(25,30)(29,36)(34,35);(5,6,12,24,10,15,8,11)(7,13,25,35,18,29,17,22)(9,20,26,23,21,14,19,33)(16,28,32,36,31,34,30,27);(5,7)(6,13)(8,17)(9,16)(10,18)(11,22)(12,25)(14,28)(15,29)(19,30)(20,34)(21,31)(23,27)(24,35)(26,32)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=di,d^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55372);