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 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,22,25)(19,21,27,23)(20,28,29,24)(26,30,32,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19,22,27)(18,23,25,21)(20,26,29,32)(24,30,28,31)(33,35,38,46)(34,39,42,43)(36,44,45,37)(40,41,47,48);(2,6)(7,10)(8,12)(14,16)(20,29)(24,28)(26,32)(30,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,29)(18,24,25,28)(19,26,27,32)(21,31,23,30)(33,36)(34,40)(35,44)(37,46)(38,45)(39,41)(42,47)(43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21)(18,19)(20,30)(22,23)(24,32)(25,27)(26,28)(29,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,31)(28,30)(29,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8732383);