Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| 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:
(1,2)(3,4)(5,6,12,18,10,16,13,27)(7,19,25,34,23,36,26,32)(8,9,17,11,28,15,30,20)(14,29,21,33,31,24,35,22);(5,7,20,24,13,26,15,33,10,23,11,29,12,25,9,22)(6,14,17,34,18,21,28,36,16,31,30,32,27,35,8,19);(2,4)(5,8)(6,15)(7,21)(9,16)(10,28)(11,27)(12,17)(13,30)(14,26)(18,20)(19,22)(23,35)(24,32)(25,31)(29,34)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,d^c=dhi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55623);