Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,19)(8,23)(9,17)(10,26,28,16)(11,15)(12,18)(13,20)(14,24,33,21)(22,31)(25,29,27,34)(30,35,32,36);(1,2,3,4)(5,7,9,20,12,8,11,22)(6,13,15,19,18,31,17,23)(10,21,25,35,28,24,27,36)(14,16,30,29,33,26,32,34);(5,8,12,7)(6,14,18,33)(9,20,11,22)(10,24,28,21)(13,29,31,34)(15,32,17,30)(16,23,26,19)(25,35,27,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^c=di,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, 55216);