Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,30)(20,25)(21,29)(22,27)(23,28)(24,33)(26,35)(31,46)(32,40)(34,45)(36,44)(37,41)(38,43)(39,47)(42,48);(1,2,4,5)(3,6,7,8)(17,19)(18,24)(20,34)(21,32)(22,35)(23,33)(25,41)(26,40)(27,29)(28,30)(31,36)(37,38)(39,44)(42,46)(43,45)(47,48);(9,11,13,12)(10,14,15,16)(17,20,35,48,23,38,21,36)(18,25,26,42,28,43,29,44)(19,31,32,37,33,47,22,34)(24,39,27,45,30,46,40,41);(1,3,4,7)(2,6,5,8)(17,21,23,35)(18,26,28,29)(19,32,33,22)(20,36,38,48)(24,40,30,27)(25,42,43,44)(31,37,47,34)(39,41,46,45);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,22,21,33,23,32,35,19)(18,27,29,24,28,40,26,30)(20,37,36,31,38,34,48,47)(25,41,44,46,43,45,42,39)Polycyclic presentation:
2^9|b^2=fi,c^2=ghi,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6744189);