Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| 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 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,18,9,17,10,16,12,19,29,36,25,15,30,31)(7,20,23,32,21,13,27,33,8,24,28,14,26,34,22,35);(1,2)(3,4)(5,7,9,21,12,8,25,26)(6,13,15,20,19,34,17,24)(10,27,29,28,30,22,11,23)(14,18,33,31,32,36,35,16);(2,4)(5,8)(6,14)(7,12)(9,21)(10,28)(11,22)(13,31)(15,35)(16,34)(17,33)(18,20)(19,32)(23,30)(24,36)(25,26)(27,29)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250344, #72250347, #72250351, … |
| maximal normal subgroup of | #72250344, #72250347, #72250351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55421);