Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1478 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)(7,17,20,27)(8,22)(9,14)(10,15)(11,16)(12,19,28,21)(13,24)(18,32,35,26)(23,29)(25,30)(31,33,36,34);(1,2)(3,4)(5,7,10,21,11,20,9,19)(6,12,14,17,16,28,15,27)(8,18,24,34,25,35,23,33)(13,26,22,31,29,32,30,36);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,22,16,29,15,30)(7,18,19,33,20,35,21,34)(12,26,17,31,28,32,27,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,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, 30863);