Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4⋊#95 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,14,18,15,19,13,29)(11,20,22,16,24,31,23,32)(12,25,27,33,28,39,26,17)(21,30,35,38,37,40,36,34);(1,2,3,6,4,5,7,8)(9,11)(10,16)(12,21)(13,23)(14,22)(15,24)(17,30)(18,31)(19,32)(20,29)(25,38)(26,36)(27,35)(28,37)(33,40)(34,39);(2,5)(6,8)(9,12,13,26,15,28,14,27)(10,17,18,25,19,33,29,39)(11,21,23,36,24,37,22,35)(16,30,31,38,32,40,20,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16359);