Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,40)(20,32,35,25)(21,26,22,27)(29,45,47,38)(30,39,36,43)(33,48,46,41)(34,44,37,42);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,30)(22,36)(23,31)(25,41)(26,39)(27,43)(28,40)(29,34)(32,48)(35,46)(37,47)(38,42)(44,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,43)(27,39)(28,40)(32,45)(33,34)(35,47)(37,46)(41,42)(44,48);(10,14)(13,16)(17,20,22,34,23,35,21,37)(18,25,27,42,28,32,26,44)(19,29,30,33,31,47,36,46)(24,38,39,41,40,45,43,48);(1,4,5,3)(2,7,8,6)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,34,35,37)(24,39,40,43)(25,42,32,44)(29,46,47,33)(38,48,45,41)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520490);