Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,13,16)(11,19,21,38)(12,23,26,37)(14,28,25,18)(15,30,31,20)(17,34,33,27)(22,39,29,35)(24,36,40,32);(9,11,13,21)(10,15,16,31)(12,24,26,40)(14,29,25,22)(17,35,33,39)(18,36,28,32)(19,37,38,23)(20,34,30,27);(1,3,4,8,5,2,6,7);(2,3)(7,8)(10,16)(15,31)(17,33)(18,28)(19,38)(23,37)(32,36)(35,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391274);