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 | #1440 |
| 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:
(2,3)(5,6,10,14,11,16,9,15)(7,17,20,28,21,32,19,12)(8,22,24,31,26,30,25,13)(18,34,35,33,36,27,23,29);(1,2)(3,4)(5,7)(6,12,16,28)(8,23,26,35)(9,19)(10,20)(11,21)(13,29)(14,17,15,32)(18,25,36,24)(22,34)(27,30)(31,33);(1,3,4,2)(5,8,9,24,11,26,10,25)(6,13,14,30,16,31,15,22)(7,18,19,35,21,36,20,23)(12,27,17,33,28,34,32,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^c=fi,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, 31002);