Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
1469 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)(17,18,21,26,22,27,33,41)(19,23,30,35,29,25,39,40)(20,24,31,37,32,38,44,48)(28,34,43,45,42,36,46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,34)(25,27)(26,35)(31,43)(32,42)(33,39)(36,38)(37,45)(40,41)(44,46)(47,48);(10,14)(11,15)(17,20,21,31,22,32,33,44)(18,24,26,37,27,38,41,48)(19,28,30,43,29,42,39,46)(23,34,35,45,25,36,40,47);(2,5)(6,8)(10,14)(13,16)(17,19)(18,25)(20,28)(21,30)(22,29)(23,27)(24,36)(26,40)(31,43)(32,42)(33,39)(34,38)(35,41)(37,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(19,29)(23,25)(28,42)(30,39)(34,36)(35,40)(43,46)(45,47)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=cg,d^b=df,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319054);