Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363286 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1564 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)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,24,29)(19,30,34,46)(20,26,37,21)(22,28,23,27)(25,41,44,33)(31,47,35,32)(36,39,40,42)(38,48,45,43);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(17,19)(18,25)(20,35)(21,38)(22,33)(23,41)(24,34)(26,45)(27,30)(28,46)(29,44)(31,37)(32,39)(36,43)(40,48)(42,47);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20)(18,26)(19,31)(21,29)(22,40)(23,36)(24,37)(25,38)(27,39)(28,42)(30,47)(32,46)(33,43)(34,35)(41,48)(44,45);(17,21,24,26)(18,20,29,37)(19,32,34,47)(22,39,23,42)(25,43,44,48)(27,36,28,40)(30,31,46,35)(33,45,41,38);(2,8)(6,14)(7,9)(13,15)(17,22,24,23)(18,27,29,28)(19,33,34,41)(20,36,37,40)(21,39,26,42)(25,30,44,46)(31,48,35,43)(32,45,47,38)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=h,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,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, 6270511);