Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #390 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(6,7)(9,10,14,18,15,19,13,27)(11,16,21,32,22,33,28,31)(12,23,25,17,26,35,29,34)(20,36,38,40,24,39,37,30);(5,6)(7,8)(9,11)(10,16)(12,24,26,20)(13,28)(14,21)(15,22)(17,30,34,40)(18,32)(19,33)(23,39,35,36)(25,37,29,38)(27,31);(2,4)(5,7,8,6)(9,12)(10,17)(11,20)(13,29)(14,25)(15,26)(16,30)(18,35)(19,34)(21,38)(22,24)(23,27)(28,37)(31,39)(32,36)(33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fi,e^c=ei,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30946);