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 | |
4252 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)(3,9)(10,14)(11,16)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(35,44)(39,41)(45,47)(46,48);(3,9)(4,8)(5,13)(10,14)(18,23)(20,29)(21,25)(26,32)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(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=hi,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, 8924188);