Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
506 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)(3,6)(4,8)(5,7)(9,10,14,19,15,12,26,28,16,20,32,38,31,29,13,18)(11,21,23,36,24,37,22,27,25,39,30,17,33,34,40,35);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,30)(14,22)(15,33)(16,25)(18,37)(19,21)(20,36)(23,26)(24,31)(28,34)(29,35)(32,40)(38,39);(1,4,5,3)(2,6,7,8);(9,12,14,28,15,20,26,38,16,29,32,18,31,10,13,19)(11,21,30,17,24,37,40,35,25,39,23,36,33,34,22,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405242);