Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62201 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,10,12,15)(4,8,6,13)(7,14,16,11)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,39)(30,45)(31,40)(33,41)(34,48)(35,43)(37,44)(38,47)(42,46);(1,2)(3,7)(4,13)(5,9)(6,8)(10,11)(12,16)(14,15)(17,19,23,31)(18,24,27,39)(20,32,22,29)(21,34,35,47)(25,40,36,28)(26,42,43,45)(30,37,46,33)(38,41,48,44);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,16)(10,13)(17,20,23,22)(18,25,27,36)(19,29,31,32)(21,33,35,37)(24,28,39,40)(26,41,43,44)(30,34,46,47)(38,45,48,42);(1,4,5,6)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,35)(18,26,27,43)(19,30,31,46)(20,33,22,37)(24,38,39,48)(25,41,36,44)(28,45,40,42)(29,34,32,47);(2,9)(7,16)(8,13)(10,15)(18,27)(19,31)(25,36)(26,43)(29,32)(30,46)(34,47)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274538);