Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1474 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)(7,15,18,11)(8,19,22,33)(9,14)(10,13)(12,21,27,36)(16,30)(17,29,32,25)(20,34)(23,28)(24,31)(26,35);(1,2)(3,4)(5,7)(6,11)(8,20,22,16)(9,17)(10,18)(12,26,27,24)(13,15)(14,25)(19,30,33,34)(21,35,36,31)(23,32)(28,29);(2,4)(5,8)(6,12,13,27)(7,16)(9,21)(10,22)(11,24,15,26)(14,19,28,33)(17,31)(18,20)(23,36)(25,34,29,30)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30956);