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 | |
4158 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(2,7)(3,11)(4,14)(13,15)(17,19)(18,24)(20,30)(21,31)(22,25)(23,27)(26,28)(29,32)(34,40)(35,36)(42,47)(43,44);(2,7)(4,14)(8,10)(12,16)(18,25)(19,27)(21,32)(28,30)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,31,24,29)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,30)(27,31)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22)(18,27)(19,25)(20,29)(21,28)(23,24)(26,31)(30,32)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8509988);