Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907945 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,26)(19,28,30,24)(20,32,33,41)(21,27,23,37)(25,36,42,34)(29,44,46,48)(31,40,35,39)(38,47,43,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,29,33,46)(21,35,23,31)(25,38,42,43)(27,40,37,39)(32,48,41,44)(34,47,36,45);(3,8)(5,7)(10,13)(11,15)(17,20,21,36,22,33,23,34)(18,25,37,41,26,42,27,32)(19,29,35,45,30,46,31,47)(24,38,39,44,28,43,40,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(20,33)(27,37)(31,35)(34,36)(38,43)(44,48);(10,13)(14,16)(17,21,22,23)(18,27,26,37)(19,31,30,35)(20,34,33,36)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6847396);