Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11087 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,11,13,17)(8,22,24,31)(9,20)(10,21)(12,27,14,28)(19,23,29,25)(26,32);(1,2,4,7)(3,8,12,19)(5,9,15,20)(6,10,16,21)(11,23,27,22)(13,24,14,29)(17,25,28,31)(18,26,30,32);(1,3)(2,8)(4,14)(5,17)(6,13)(7,19)(9,25)(10,24)(11,18)(12,16)(15,27)(20,31)(21,29)(22,32)(23,26)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280674, #72282255, #72282782, … |
| maximal normal subgroup of | #72280674, #72282255, #72282782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55436);