Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
686 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)(5,6,9,16,10,12,25,27,11,15,23,32,8,14,22,31)(7,17,20,34,18,13,28,30,21,26,24,36,19,29,35,33);(6,12,15,14)(9,23)(13,17,29,26)(16,31,32,27)(20,24)(22,25)(28,35)(30,36,33,34);(2,4)(5,7)(6,13,15,29)(8,19)(9,24)(10,18)(11,21)(12,26,14,17)(16,33,32,30)(20,23)(22,28)(25,35)(27,34,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72231475, #72231501, #72231517, … |
| maximal normal subgroup of | #72231475, #72231501, #72231517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56619);