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 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,5)(4,8,14,16)(10,15,12,13)(17,18,22,25)(19,26,20,23)(21,24,30,29)(27,32,28,31)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,44,47,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,38,45)(34,39,42,40)(36,43,41,47)(37,44,46,48);(1,3,6,11)(2,7,9,5)(4,12,14,10)(8,13,16,15)(17,18,22,25)(19,23,20,26)(21,29,30,24)(27,32,28,31)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,48,46,44);(1,3)(2,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,44,40,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35)(34,40)(36,47)(37,44)(38,45)(39,42)(41,43)(46,48);(1,5,6,7)(2,3,9,11)(4,13,14,15)(8,10,16,12)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37)(34,41)(35,44)(36,42)(38,46)(39,47)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,e^2=gh,f^2=i,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, 8934539);