Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2076 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,14,10,26)(7,16,19,29,17,32,20,35)(8,21,23,30,25,36,24,13)(12,22,28,18,27,34,31,33);(1,2)(3,4)(12,27)(16,32)(28,31)(29,35);(2,4)(5,7,10,19,11,17,9,20)(6,12,15,31,14,27,26,28)(8,22,24,18,25,34,23,33)(13,29,21,16,30,35,36,32)Polycyclic presentation:
2^9|a^2=g,c^2=dh,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^c=di,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257135, #72257138, #72257143, … |
| maximal normal subgroup of | #72257135, #72257138, #72257143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55828);