Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2550 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)(5,6,7,8)(9,10)(11,20)(12,24,28,35)(13,29)(14,18)(15,19)(16,22)(17,27,34,26)(21,37,25,31)(23,32)(30,33)(36,38,40,39);(2,4)(9,11)(10,16)(12,25,28,21)(13,30)(14,22)(15,23)(17,31,34,37)(18,32)(19,33)(20,29)(24,36,35,40)(26,38,27,39);(9,12)(10,17)(11,21)(13,27)(14,26)(15,28)(16,31)(18,35)(19,34)(20,36)(22,39)(23,25)(24,29)(30,38)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,f^b=fi,d^c=di,e^c=ei,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30826);