Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359226 |
| 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 | #62471 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,18,24,29)(19,30,35,43)(20,21,26,39)(22,28,23,27)(25,33,44,34)(31,46,45,32)(36,41,40,38)(37,47,48,42);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19,24,35)(18,25,29,44)(20,31,26,45)(21,37,39,48)(22,34,23,33)(27,30,28,43)(32,41,46,38)(36,47,40,42);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,24,26)(18,21,29,39)(19,31,35,45)(22,40,23,36)(25,37,44,48)(27,41,28,38)(30,46,43,32)(33,47,34,42);(17,21)(18,26)(19,32)(20,29)(22,38)(23,41)(24,39)(25,42)(27,40)(28,36)(30,31)(33,37)(34,48)(35,46)(43,45)(44,47);(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,36,26,40)(21,38,39,41)(25,43,44,30)(31,42,45,47)(32,37,46,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^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=eh,f^b=fh,e^c=eh,g^c=gi,f^d=fh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266447);