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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,8,13,12)(10,15,16,14)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(2,4)(7,9)(8,12)(14,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(5,13)(8,12)(10,16)(14,15)(18,21)(23,25)(24,28)(31,32)(34,38)(39,41)(40,44)(46,47);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134506);