Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501527 |
| 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×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,25,28,22,31,11,12,23,14,10,16,8,21)(7,17,18,33,26,29,36,27,19,34,35,30,20,32,24,13);(1,2)(3,4)(6,12)(8,22)(9,23)(13,27)(16,28)(18,35)(24,36)(30,33);(2,4)(5,7)(6,13)(8,18)(9,24)(10,26)(11,19)(12,27)(14,29)(15,32)(16,33)(17,21)(20,25)(22,35)(23,36)(28,30)(31,34);(5,8,10,23,11,22,25,9)(6,14,28,21,12,15,16,31)(7,18,26,36,19,35,20,24)(13,29,30,17,27,32,33,34)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,d^b=di,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, 408723);