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 | |
4224 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)(3,11,12,8)(4,9,15,14)(5,13,16,10)(18,23)(19,26)(25,31)(27,29)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(2,7)(4,15)(8,11)(10,13)(17,18,22,23)(19,24,26,21)(20,25,28,31)(27,32,29,30)(35,44)(39,41)(45,47)(46,48);(2,7)(3,12)(4,15)(5,16)(19,26)(20,28)(21,24)(25,31)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,16)(13,15)(17,19)(18,24)(20,29)(21,23)(22,26)(25,30)(27,28)(31,32)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,4)(2,9)(3,13)(5,8)(6,15)(7,14)(10,12)(11,16)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,5)(2,10)(3,14)(4,8)(6,16)(7,13)(9,12)(11,15)(17,21,22,24)(18,19,23,26)(20,30,28,32)(25,27,31,29)(33,37,38,40)(34,42,43,36)(35,46,44,48)(39,47,41,45)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=i,e^2=ghi,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, 8924190);