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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,13,15,9)(11,16,14,12)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(1,3,6,8)(2,7,10,4)(5,14,15,11)(9,12,13,16)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4)(2,8)(3,10)(5,12)(6,7)(9,14)(11,13)(15,16)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,15)(2,9,10,13)(3,11,8,14)(4,12,7,16)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900766);