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 | |
4136 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,23,25)(19,24,21,27)(20,26,31,30)(22,29,32,28)(33,34)(35,37)(36,45)(38,46)(39,41)(40,44)(42,43)(47,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19)(18,24)(20,29)(21,23)(22,30)(25,27)(26,32)(28,31)(33,35,39,44)(34,40,41,37)(36,42,46,48)(38,43,45,47);(2,7)(3,10)(8,15)(11,12)(18,25)(19,21)(20,31)(22,32)(33,36,39,46)(34,38,41,45)(35,42,44,48)(37,47,40,43);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27)(33,36)(34,38)(35,42)(37,47)(39,46)(40,43)(41,45)(44,48);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30)(33,37,39,40)(34,35,41,44)(36,43,46,47)(38,48,45,42);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)(33,38)(34,36)(35,43)(37,48)(39,45)(40,42)(41,46)(44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9405631);