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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(34,39)(35,42)(37,45)(44,47);(2,7)(4,9)(8,13)(12,14)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,7)(3,10)(8,13)(11,16)(18,24)(19,27)(20,29)(26,30)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10049326);