Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,19,29)(8,13,21,11)(9,14)(10,15)(12,18,27,22)(17,32)(20,28,36,25)(23,30)(24,33)(26,35)(31,34);(1,2,3,4)(6,11,15,13)(12,24,27,26)(14,25,30,28)(16,31,29,32);(5,7)(6,12)(8,17)(9,22)(10,19)(11,24)(13,26)(14,29)(15,27)(16,30)(18,23)(20,35)(21,34)(25,32)(28,31)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,f^b=fhi,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53683);