Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6,5,7,3,8)(9,10,14,18,15,19,13,29)(11,20,24,31,23,35,22,16)(12,25,27,33,28,17,26,32)(21,34,38,30,37,40,36,39);(2,6)(3,4)(7,8)(9,11)(10,16,19,31)(12,21)(13,22)(14,24)(15,23)(17,30,25,39)(18,20,29,35)(26,36)(27,38)(28,37)(32,40,33,34);(2,6)(3,4)(7,8)(9,12,15,28)(10,17,19,25)(11,21,23,37)(13,27,14,26)(16,30,31,39)(18,33,29,32)(20,34,35,40)(22,38,24,36);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280251 |
| maximal normal subgroup of | #72280251 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106273);