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 | |
4129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,15)(11,12)(33,34)(35,42)(36,37)(38,39)(40,45)(41,48)(43,44)(46,47);(1,2,6,7)(3,8,10,5)(4,9,14,15)(11,16,12,13)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(34,39)(36,41)(40,42)(43,46);(3,10)(5,8)(11,12)(13,16)(18,23)(19,26)(21,30)(27,29)(33,35,38,45)(34,40,39,42)(36,43,41,46)(37,47,48,44);(1,3)(2,8)(4,12)(5,7)(6,10)(9,13)(11,14)(15,16)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,15)(8,16)(10,12)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36,38,41)(34,37,39,48)(35,43,45,46)(40,47,42,44);(1,5,6,8)(2,3,7,10)(4,13,14,16)(9,11,15,12)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37)(34,41)(35,44)(36,39)(38,48)(40,43)(42,46)(45,47)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826589);