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 | #2668 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,27)(19,23,30,32)(20,25,29,24)(21,26,28,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,2,6,7)(3,8,11,16)(4,9,14,13)(5,10,15,12)(17,19,22,30)(18,23,27,32)(20,28,29,21)(24,26,25,31)(34,39)(36,40)(41,45)(44,47);(2,7)(3,11)(4,14)(5,15)(17,20,22,29)(18,24,27,25)(19,21,30,28)(23,31,32,26)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,3)(2,8)(4,10)(5,13)(6,11)(7,16)(9,15)(12,14)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,27)(26,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4)(2,9)(3,12)(5,8)(6,14)(7,13)(10,11)(15,16)(17,21,22,28)(18,26,27,31)(19,29,30,20)(23,24,32,25)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5)(2,10)(3,13)(4,8)(6,15)(7,12)(9,11)(14,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8884531);