Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #44760 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,5,8,7)(18,23)(21,34)(24,28)(25,36)(30,43)(33,35)(38,40)(45,46);(1,3)(2,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(20,31)(26,37)(30,43)(33,35)(38,40)(47,48);(9,11)(10,13)(12,16)(14,15)(17,18,22,23)(19,28,27,24)(20,25,31,36)(21,26,34,37)(29,40,32,38)(30,44,43,39)(33,41,35,42)(45,48,46,47);(2,4)(3,8)(10,14)(13,15)(17,19,22,27)(18,24,23,28)(20,32,31,29)(21,30,34,43)(25,40,36,38)(26,39,37,44)(33,46,35,45)(41,48,42,47);(1,4,6,2)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,20,22,31)(18,25,23,36)(19,29,27,32)(21,35,34,33)(24,38,28,40)(26,42,37,41)(30,46,43,45)(39,48,44,47);(1,5)(2,8)(3,4)(6,7)(9,11)(10,15)(12,16)(13,14)(17,21)(18,26)(19,30)(20,33)(22,34)(23,37)(24,39)(25,41)(27,43)(28,44)(29,45)(31,35)(32,46)(36,42)(38,47)(40,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126682);