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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(9,14)(10,15)(11,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(18,19)(23,25)(24,26)(30,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(3,8)(9,14)(10,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,11,12,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022764);