Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1438 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)(7,15,17,11)(8,19)(9,14)(10,13)(12,20)(16,23,29,27)(18,28,31,24)(21,32)(22,26)(25,33)(30,34,36,35);(1,2)(3,4)(5,7,9,18,10,17,22,31)(6,11,14,24,13,15,26,28)(8,16,21,30,20,29,33,36)(12,23,25,34,19,27,32,35);(2,4)(5,8)(6,12,13,19)(7,16)(9,21)(10,20)(11,23,15,27)(14,25,26,32)(17,29)(18,30)(22,33)(24,34,28,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30917);