Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13316 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,22,32,35,36,10,13,26,28,11,16,19,33)(7,17,8,21,23,14,27,29,18,34,24,30,20,31,25,12);(1,2)(3,4)(5,7)(6,12)(8,19)(9,25)(10,18)(11,23)(13,29)(14,28)(15,31)(16,21)(17,33)(20,22)(24,35)(26,27)(30,32)(34,36);(2,4)(6,13)(7,18)(9,26)(14,31)(16,32)(17,34)(19,35)(20,23);(5,8,22,25,10,24,11,27)(6,14,16,34,13,31,32,17)(7,19,20,9,18,35,23,26)(12,28,21,36,29,15,30,33)Polycyclic presentation:
2^9|a^2=e,d^2=fhi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,e^c=ei,e^d=ehi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72250721, #72252778, #72252788 |
| maximal normal subgroup of | #72250721, #72252778, #72252788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411673);