Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,22,26,33,40)(19,23,29,36,28,35,42,46)(20,24,32,39,31,38,44,48)(27,34,41,45,30,37,43,47);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,30)(21,29)(22,28)(24,37)(25,36)(26,35)(27,31)(32,43)(33,42)(34,38)(39,47)(40,46)(41,44)(45,48);(3,8)(5,7)(10,13)(14,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,39)(26,38)(28,30)(29,41)(33,44)(35,37)(36,45)(40,48)(42,43)(46,47);(2,6)(5,7)(19,28)(23,35)(27,30)(29,42)(34,37)(36,46)(41,43)(45,47);(9,11,12,15)(10,14,13,16)(20,31)(24,38)(27,30)(32,44)(34,37)(39,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^b=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282479);