Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(10,15)(11,19)(12,23)(14,26)(17,31)(18,34)(21,38)(22,40)(24,28)(25,29)(27,37)(30,32)(33,35)(36,39);(2,4)(5,6)(7,8)(9,10)(11,20)(12,24)(13,17)(14,28)(15,23)(16,21)(18,26)(19,36)(22,33)(25,39)(27,34)(29,35)(30,38)(31,37)(32,40);(6,8)(9,11,13,21)(10,16,17,20)(12,25,27,40)(14,29,26,22)(15,30,31,36)(18,35,28,33)(19,37,38,23)(24,32,34,39);(10,17)(15,31)(16,20)(18,28)(24,34)(30,36)(32,39)(33,35)Polycyclic presentation:
2^9|c^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319972, #72320788, #72320798, … |
| maximal normal subgroup of | #72319972, #72320788, #72320798, … |
| maximal quotient of | #72322951, #72322963, #72322973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406043);