Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21,22,35)(12,18)(13,28)(14,30)(15,20)(16,23,31,36)(17,26)(19,29)(24,39,37,33)(25,40,38,32)(27,34);(9,11,13,24,29,38,27,36,15,22,26,37,14,25,12,23)(10,16,18,32,30,39,17,21,20,31,34,40,19,33,28,35);(11,22)(16,31)(21,35)(23,36)(24,37)(25,38)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=ci,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33852);