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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,11,5)(4,8,12,14)(10,15,16,13)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(2,7)(5,9)(8,14)(13,15)(20,28)(24,31)(26,32)(29,30)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(4,12)(8,14)(10,16)(13,15)(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,16)(14,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,4,6,12)(2,8,7,14)(3,10,11,16)(5,13,9,15)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,5)(2,3)(4,13)(6,9)(7,11)(8,10)(12,15)(14,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150971);