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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3879 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)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18,23,20)(19,24,27,21)(22,25,30,28)(26,31,32,29)(33,34)(35,42)(36,46)(37,45)(38,40)(39,41)(43,47)(44,48);(1,2)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,21)(22,26)(23,27)(25,31)(28,29)(30,32)(33,35,40,45)(34,37,38,42)(36,43,39,44)(41,47,46,48);(2,4)(7,9)(8,12)(15,16)(17,18,23,20)(19,21,27,24)(22,25,30,28)(26,29,32,31)(33,36,40,39)(34,41,38,46)(35,43,45,44)(37,47,42,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20,23,18)(19,21,27,24)(22,25,30,28)(26,31,32,29)(33,37)(34,35)(36,47)(38,45)(39,48)(40,42)(41,43)(44,46);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,19)(20,27)(22,29)(23,24)(25,26)(28,32)(30,31)(33,38,40,34)(35,37,45,42)(36,41,39,46)(43,48,44,47);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,30)(18,25,20,28)(19,26,27,32)(21,29,24,31)(33,39,40,36)(34,41,38,46)(35,44,45,43)(37,47,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8556208);