Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2058 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
494 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)(7,16,18,35)(8,21)(9,25)(10,28)(11,15)(12,19,30,17)(13,22)(14,24)(20,36,29,32)(23,31,27,34)(26,33);(1,2,4,3)(7,17,18,19)(8,22)(9,26)(10,24)(12,16,30,35)(13,21)(14,28)(20,23,29,27)(25,33)(31,36,34,32);(1,3,4,2)(5,7,10,29,11,18,24,20)(6,12,14,34,15,30,28,31)(8,23,22,19,9,27,26,17)(13,32,21,16,33,36,25,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250159, #72250178, #72250219, … |
| maximal normal subgroup of | #72250159, #72250178, #72250219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55979);