Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1432 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)(3,7)(4,5)(6,8)(11,15)(14,16)(17,18,23,38,22,27,21,26)(19,28,37,24,31,44,30,40)(20,32,34,43,36,25,35,42)(29,46,47,41,33,45,48,39);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,40)(20,33,36,29)(21,37,23,30)(25,41,32,39)(26,44,38,28)(34,47,35,48)(42,46,43,45);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,22,36)(18,25,27,32)(19,29,31,33)(21,35,23,34)(24,39,40,41)(26,43,38,42)(28,45,44,46)(30,48,37,47);(10,13)(14,16)(17,21,22,23)(18,26,27,38)(19,30,31,37)(20,34,36,35)(24,28,40,44)(25,42,32,43)(29,47,33,48)(39,46,41,45);(9,11,12,15)(10,14,13,16)(18,27)(24,40)(25,32)(26,38)(28,44)(39,41)(42,43)(45,46)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402143);