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 | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 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)(17,18,22,26,21,25,34,41)(19,23,29,37,28,36,43,47)(20,30,33,44,32,24,38,40)(27,39,42,48,31,35,45,46);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,35)(25,36)(26,37)(30,39)(31,32)(33,42)(34,43)(38,45)(40,46)(41,47)(44,48);(10,14)(11,15)(17,20,22,33,21,32,34,38)(18,24,26,40,25,30,41,44)(19,27,29,42,28,31,43,45)(23,35,37,46,36,39,47,48);(2,5)(6,8)(10,14)(13,16)(17,19)(18,23)(20,31)(21,28)(22,29)(24,39)(25,36)(26,37)(27,32)(30,35)(33,45)(34,43)(38,42)(40,48)(41,47)(44,46);(1,3,4,7)(2,6,5,8)(19,28)(23,36)(27,31)(29,43)(35,39)(37,47)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,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, 6319063);