Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
688 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)(6,13)(7,18,22,33)(8,11)(9,25)(10,24)(14,19,32,35)(15,27)(17,28)(20,34,23,21)(29,31,30,36);(5,6,8,13,12,17,11,28)(7,19,20,31,22,35,23,36)(9,26,10,27,24,16,25,15)(14,18,30,21,32,33,29,34);(2,4)(5,7)(6,14)(8,23)(9,21)(10,18)(11,20)(12,22)(13,29)(15,19)(16,31)(17,32)(24,34)(25,33)(26,36)(27,35)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250750, #72250752, #72250768, … |
| maximal normal subgroup of | #72250750, #72250752, #72250768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55347);