Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501524 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,28,31,9,24,11,15,29,32,12,16,25,33)(7,8,18,22,34,30,35,36,19,13,26,21,20,23,17,27);(1,2)(3,4)(5,7)(6,13)(8,15)(9,17)(10,26)(11,19)(12,20)(14,22)(16,30)(18,29)(21,32)(23,31)(24,36)(25,35)(27,33)(28,34);(2,4)(5,8)(6,7)(9,21)(10,27)(11,13)(12,30)(14,17)(15,19)(16,34)(18,33)(20,31)(22,25)(23,28)(24,26)(29,36)(32,35);(5,9,12,10,11,25,28,29)(6,14,31,24,15,32,16,33)(7,17,20,26,19,35,34,18)(8,21,30,27,13,22,23,36)Polycyclic presentation:
2^9|a^2=f,d^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fh,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408720);