Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
1576 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,33,40)(19,23,29,36,28,35,43,47)(20,24,32,39,30,37,41,45)(27,34,42,46,31,38,44,48);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(20,30)(24,37)(27,31)(32,41)(34,38)(39,45)(42,44)(46,48);(10,14)(13,16)(17,19,21,29,22,28,33,43)(18,23,25,36,26,35,40,47)(20,31,32,44,30,27,41,42)(24,38,39,48,37,34,45,46);(1,4,6,5)(2,7,3,8)(17,20)(18,24)(19,27)(21,32)(22,30)(23,34)(25,39)(26,37)(28,31)(29,42)(33,41)(35,38)(36,46)(40,45)(43,44)(47,48);(1,5,6,4)(2,7,3,8)(19,28)(20,30)(23,35)(24,37)(29,43)(32,41)(36,47)(39,45)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300524);