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 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(10,14)(11,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19,23,31)(18,24,22,30)(20,28,29,21)(25,27,26,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(3,9)(4,8)(5,13)(10,14)(17,20,23,29)(18,25,22,26)(19,21,31,28)(24,32,30,27)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20)(18,26)(19,28)(21,31)(22,25)(23,29)(24,32)(27,30)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,28)(18,27,22,32)(19,29,31,20)(24,25,30,26)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,22,23,18)(19,30,31,24)(20,25,29,26)(21,32,28,27)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=hi,e^2=gi,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, 8922576);