Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137786 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1899 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11043 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,14)(4,18)(5,12)(6,22,23,11)(7,13)(8,15)(9,20,27,16)(10,19)(17,26)(21,24,32,28)(25,30,29,31);(1,3,5,15,7,4,19,17)(2,8,10,14,13,26,12,18)(6,16,21,30,23,20,32,31)(9,11,25,24,27,22,29,28);(1,4)(2,9)(3,7)(5,17)(6,20)(8,24)(10,25)(11,18)(12,29)(13,27)(14,22)(15,19)(16,23)(21,31)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,f^c=fi,g^c=gh,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55213);