Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1386 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)(17,18,23,28)(19,26,31,44)(20,25,33,40)(21,36,32,24)(22,38,29,27)(30,43,46,41)(34,39,37,48)(35,47,45,42);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,29,21,23,31,22,32)(18,24,38,44,28,36,27,26)(20,30,45,37,33,46,35,34)(25,39,47,41,40,48,42,43);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20,23,33)(18,25,28,40)(19,30,31,46)(21,37,32,34)(22,35,29,45)(24,39,36,48)(26,43,44,41)(27,42,38,47);(2,6)(5,7)(10,14)(13,16)(19,31)(20,33)(21,32)(24,36)(25,40)(26,44)(35,45)(42,47);(2,6)(5,7)(17,21,29,31,23,32,22,19)(18,26,38,24,28,44,27,36)(20,34,45,30,33,37,35,46)(25,41,47,48,40,43,42,39)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507702);