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 | |
4191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,4)(5,9,14,15)(11,13,16,12)(33,34,38,37)(35,41,43,39)(36,44,46,40)(42,47,45,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,13)(12,15)(14,16)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(2,8)(3,10)(9,15)(11,16)(18,23)(20,29)(24,31)(26,32)(33,35,38,43)(34,39,37,41)(36,42,46,45)(40,47,44,48);(1,4)(2,3)(5,13)(6,7)(8,10)(9,16)(11,15)(12,14)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45);(1,2)(3,4)(5,9)(6,8)(7,10)(11,12)(13,16)(14,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,48)(45,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37,38,34)(35,41,43,39)(36,40,46,44)(42,47,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540719);