Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36652 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,22,27,30,44)(19,24,21,36,31,40,38,26)(20,32,35,48,34,25,41,43)(29,46,37,45,47,39,33,42);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,23)(22,31)(25,39)(26,44)(27,40)(28,36)(30,38)(32,46)(33,41)(34,47)(35,37)(42,43)(45,48);(10,14)(11,15)(17,20,23,35,22,34,30,41)(18,25,28,43,27,32,44,48)(19,29,21,37,31,47,38,33)(24,39,36,42,40,46,26,45);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21)(18,26)(19,30)(20,33)(22,38)(23,31)(24,28)(25,42)(27,36)(29,35)(32,45)(34,37)(39,48)(40,44)(41,47)(43,46);(2,5)(6,8)(18,27)(19,31)(21,38)(25,32)(28,44)(29,47)(33,37)(43,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6318809);