Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2071 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,18,13,20,31,19)(7,8,24,28,25,30,35,29)(9,17,10,16,11,26,33,36)(14,22,34,21,32,15,27,23);(1,2)(3,4)(5,7)(6,14)(8,26)(9,22)(10,23)(11,15)(12,35)(13,25)(16,28)(17,30)(18,27)(19,34)(20,32)(21,33)(24,31)(29,36);(2,4)(5,8,9,27,12,29,10,32,13,30,11,34,31,28,33,14)(6,15,16,25,19,23,17,35,20,22,36,7,18,21,26,24)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250044, #72250050, #72250053, … |
| maximal normal subgroup of | #72250044, #72250050, #72250053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55767);