Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,31)(20,35)(21,25)(22,29)(23,28)(24,30)(26,39)(27,34)(32,47)(33,42)(36,43)(37,41)(38,45)(40,44)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,33,24,34,22,21)(18,25,29,27,30,42,28,31)(20,36,39,37,40,48,38,32)(26,43,35,47,45,46,44,41);(3,7)(6,8)(11,15)(13,16)(17,20,22,39,24,40,23,38)(18,26,28,35,30,45,29,44)(19,32,21,36,34,37,33,48)(25,41,31,43,42,47,27,46);(1,3,4,7)(2,6,5,8)(17,21,22,34,24,33,23,19)(18,27,28,25,30,31,29,42)(20,37,38,36,40,32,39,48)(26,41,44,46,45,47,35,43)Polycyclic presentation:
2^9|a^2=eg,b^2=hi,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377998);