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 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,8)(9,14)(10,15)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,47)(45,46);(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(35,42)(39,41)(43,44)(47,48);(3,8)(4,11)(5,13)(9,14)(18,23)(20,29)(21,25)(26,32)(33,35,38,42)(34,39,37,41)(36,44,46,43)(40,48,45,47);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,36,38,46)(34,40,37,45)(35,43,42,44)(39,47,41,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,35)(34,41)(36,44)(37,39)(38,42)(40,47)(43,46)(45,48);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,12,11,16)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,37,38,34)(35,39,42,41)(36,45,46,40)(43,47,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=i,d^2=i,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, 8900290);