Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #380 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1598 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)(6,8)(9,10,13,18,15,20,14,19)(11,21,23,31,24,35,28,16)(12,17,26,33,27,34,29,36)(22,37,38,32,25,40,39,30);(2,4)(9,11,14,23,15,24,13,28)(10,16,19,21,20,31,18,35)(12,25,26,38,27,22,29,39)(17,32,33,37,34,30,36,40);(5,6,7,8)(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,30)(18,33)(19,36)(20,34)(21,37)(23,38)(24,25)(28,39)(31,32)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fi,g^b=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31227);