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 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(34,38)(35,43)(36,46)(42,48);(1,2,6,3)(4,7,12,9)(5,10,13,8)(11,15,16,14)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(2,3)(7,9)(8,10)(14,15)(18,20)(24,25)(26,28)(31,32)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662346);