Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1360 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,29)(19,24,20,33)(21,36,39,45)(22,27,34,28)(25,32,42,31)(26,43,46,38)(30,41,35,44)(37,40,47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,23)(21,37)(22,31)(25,27)(26,41)(28,42)(29,33)(30,38)(32,34)(35,43)(36,40)(39,47)(44,46)(45,48);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,34,32,23,19,22,31)(18,25,27,24,29,42,28,33)(21,30,43,47,39,35,38,37)(26,44,45,48,46,41,36,40);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,21,22,38,23,39,34,43)(18,26,28,36,29,46,27,45)(19,30,32,37,20,35,31,47)(24,40,25,41,33,48,42,44);(2,8)(6,7)(17,22,23,34)(18,27,29,28)(19,31,20,32)(21,38,39,43)(24,25,33,42)(26,45,46,36)(30,47,35,37)(40,41,48,44)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fhi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510938);