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 | #466 | 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 | |
419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,21,24)(19,27,29,46)(20,31,34,45)(22,36,33,26)(23,38,39,28)(25,42,41,35)(30,47,37,43)(32,44,48,40);(17,19,21,29)(18,23,24,39)(20,32,34,48)(22,37,33,30)(25,43,41,47)(26,44,36,40)(27,45,46,31)(28,42,38,35);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16);(3,9)(6,12)(10,15)(13,16)(18,24)(23,39)(25,41)(26,36)(27,46)(31,45)(40,44)(43,47)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=di,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, 391225);