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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,8)(5,13,15,10)(11,14,16,12)(17,18)(19,24)(20,25)(21,28)(22,23)(26,32)(27,30)(29,31);(2,7)(4,8)(10,13)(12,14)(17,18)(19,24)(20,29)(21,26)(22,23)(25,31)(27,30)(28,32)(33,34,38,42)(35,40,45,39)(36,37,41,47)(43,48,46,44);(2,7)(3,9)(10,13)(11,16)(18,23)(19,27)(25,29)(26,28)(33,35,38,45)(34,39,42,40)(36,43,41,46)(37,44,47,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,13)(15,16)(17,19,22,27)(18,24,23,30)(20,26,31,28)(21,25,32,29)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,27)(30,32)(33,36)(34,37)(35,43)(38,41)(39,44)(40,48)(42,47)(45,46);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,21)(18,26)(19,25)(20,30)(22,32)(23,28)(24,31)(27,29)(33,37)(34,41)(35,44)(36,42)(38,47)(39,46)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,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^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050540);