Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | |
1510 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)(17,18,22,27)(19,28,23,24)(20,31,33,39)(21,35,37,26)(25,29,42,34)(30,40,38,44)(32,48,36,47)(41,46,43,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,30,37,38)(25,39,42,31)(26,40,35,44)(32,45,36,46)(41,48,43,47);(3,8)(5,7)(10,13)(11,15)(17,20,23,34,22,33,19,29)(18,25,28,31,27,42,24,39)(21,36,38,45,37,32,30,46)(26,43,44,48,35,41,40,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,37)(18,26,27,35)(19,30,23,38)(20,32,33,36)(24,40,28,44)(25,41,42,43)(29,45,34,46)(31,47,39,48);(10,13)(14,16)(20,33)(21,37)(25,42)(26,35)(29,34)(30,38)(31,39)(40,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6847293);