Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,9,15,12,16,11,10)(7,17,19,21,22,30,20,13)(8,23,26,27,28,31,25,14)(18,32,34,24,35,29,33,36);(5,7,9,20,12,22,11,19)(6,13,15,30,16,21,10,17)(8,24,26,32,28,36,25,29)(14,18,23,33,27,35,31,34);(5,8,9,25,12,28,11,26)(6,14,15,31,16,27,10,23)(7,18,19,33,22,35,20,34)(13,29,17,24,21,32,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321080, #72321110, #72321165, … |
| maximal normal subgroup of | #72321080, #72321110, #72321165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56139);