Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
844 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)(3,7)(4,5)(6,8)(9,10,13,19,14,20,28,37,15,17,29,34,12,18,22,31)(11,21,24,38,23,32,39,36,25,35,27,40,26,16,30,33);(2,5)(6,7)(9,11,22,39,12,26,29,24,15,25,28,30,14,23,13,27)(10,16,31,38,18,35,34,33,17,32,37,40,20,21,19,36);(1,3,4,8)(2,6,5,7);(10,17)(16,32)(18,20)(19,34)(21,35)(31,37)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=ei,g^b=gi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402527);