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 | |
4142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,20)(21,29)(22,24)(23,25)(26,30)(27,31)(28,32)(33,34,39,43)(35,40,38,48)(36,37,42,41)(44,46,45,47);(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35,39,38)(34,40,43,48)(36,45,42,44)(37,47,41,46);(2,4)(3,9)(5,13)(14,16)(18,24)(20,23)(21,30)(27,32)(33,36,39,42)(34,41,43,37)(35,44,38,45)(40,47,48,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,36)(34,37)(35,45)(38,44)(39,42)(40,47)(41,43)(46,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,30)(31,32)(33,37)(34,42)(35,46)(36,43)(38,47)(39,41)(40,45)(44,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,38,39,35)(34,40,43,48)(36,44,42,45)(37,47,41,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=gh,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, 8912399);