Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,15,9,14)(7,17,20,28,21,12,19,29)(8,22,24,31,26,32,25,13)(18,34,35,27,36,30,23,33);(1,2)(3,4)(5,7,11,21)(6,12,15,17)(8,23,26,35)(9,20,10,19)(13,30,31,34)(14,29,16,28)(18,24,36,25)(22,27,32,33);(2,4)(5,8)(6,13,15,31)(7,18)(9,24)(10,25)(11,26)(12,27,17,33)(14,22,16,32)(19,35)(20,23)(21,36)(28,30,29,34);(6,14,15,16)(12,28,17,29)(13,22,31,32)(27,30,33,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290935);