Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 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)(5,6,9,15,11,16,10,14)(7,17,19,12,20,29,21,28)(8,22,24,31,25,13,26,32)(18,34,35,30,23,33,36,27);(5,7,10,21,11,20,9,19)(6,12,15,29,16,28,14,17)(8,23,24,36,25,18,26,35)(13,30,31,34,22,27,32,33);(2,4)(5,8,9,24,11,25,10,26)(6,13,14,31,16,22,15,32)(7,18,19,35,20,23,21,36)(12,27,17,33,28,30,29,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30971);