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 | #34636 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1587 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)(11,13)(12,14)(15,16)(17,18,21,26,22,25,32,38)(19,23,29,35,28,34,41,45)(20,24,31,36,30,37,42,46)(27,33,40,44,39,43,47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,34)(26,35)(30,39)(31,40)(32,41)(36,44)(37,43)(38,45)(42,47)(46,48);(10,14)(11,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,39)(29,40)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(2,5)(6,8)(10,14)(13,16)(18,25)(20,30)(23,34)(26,38)(27,39)(31,42)(35,45)(40,47);(1,3,4,7)(2,6,5,8)(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48)Polycyclic presentation:
2^9|a^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315516);