Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119801 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6,4,7)(5,8)(9,10)(11,19,21,15)(12,23)(13,27)(14,17)(16,25)(18,30)(20,37,38,31)(22,32,28,35)(24,33,39,40)(26,34)(29,36);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,24)(13,28)(14,21)(16,33)(17,19)(18,35)(20,26)(22,30)(23,40)(25,39)(27,32)(29,38)(31,34)(36,37);(1,4,5,3)(2,7)(6,8)(9,12)(10,16,17,23)(11,20)(13,29)(14,25)(15,31,19,37)(18,36,27,34)(21,38)(22,39)(24,28)(26,30)(32,40,35,33)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fhi,g^b=gh,e^c=eh,f^c=fhi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26998);