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 | |
4076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(17,18,23,25)(19,24,21,27)(20,26,31,30)(22,29,32,28);(2,7)(4,14)(8,11)(10,15)(17,19)(18,24)(20,29)(21,23)(22,30)(25,27)(26,32)(28,31)(33,34,38,39)(35,40,43,48)(36,41,46,45)(37,42,47,44);(2,7)(3,12)(4,14)(5,16)(18,25)(19,21)(20,31)(22,32)(34,39)(35,43)(36,46)(37,47);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27)(33,35)(34,40)(36,42)(37,45)(38,43)(39,48)(41,47)(44,46);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30)(33,36)(34,41)(35,44)(37,40)(38,46)(39,45)(42,43)(47,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)(33,37)(34,42)(35,45)(36,40)(38,47)(39,44)(41,43)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=h,e^2=gh,f^2=h,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, 8886946);