Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29,24,30,22,28)(19,31,34,45,35,44,33,25)(20,36,38,47,39,26,41,43)(21,27,32,46,40,42,37,48);(9,10)(11,15)(12,14)(13,16)(17,19,24,35)(18,25,30,45)(20,37,39,32)(21,38,40,41)(22,34,23,33)(26,27,36,42)(28,31,29,44)(43,48,47,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,24,39)(18,26,30,36)(19,32,35,37)(21,34,40,33)(22,41,23,38)(25,42,45,27)(28,47,29,43)(31,48,44,46);(11,16)(13,15)(17,21,22,37,24,40,23,32)(18,27,28,48,30,42,29,46)(19,20,33,41,35,39,34,38)(25,43,44,26,45,47,31,36);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,38,39,41)(21,32,40,37)(25,44,45,31)(26,43,36,47)(27,46,42,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6993493);