Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365146 |
| 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 | #62529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1670 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,28)(19,29,34,40)(20,25,36,43)(21,27,22,26)(24,32,41,33)(30,45,48,39)(31,46,47,38)(35,44,37,42);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,23,34)(18,24,28,41)(20,30,36,48)(21,33,22,32)(25,38,43,46)(26,29,27,40)(31,37,47,35)(39,44,45,42);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,44)(27,42)(28,43)(29,45)(31,32)(33,47)(34,48)(39,40)(41,46);(17,20)(18,25)(19,31)(21,35)(22,37)(23,36)(24,39)(26,42)(27,44)(28,43)(29,46)(30,32)(33,48)(34,47)(38,40)(41,45);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,26,28,27)(19,32,34,33)(20,35,36,37)(24,40,41,29)(25,42,43,44)(30,47,48,31)(38,45,46,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272351);