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 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,21)(19,27,28,24)(20,26,22,25)(29,31,30,32)(35,37)(39,41)(43,45)(47,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(18,21)(19,28)(25,26)(29,30)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(2,6)(7,10)(8,12)(14,16)(17,19,23,28)(18,24,21,27)(20,29,22,30)(25,31,26,32)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,22)(18,25,21,26)(19,29,28,30)(24,31,27,32);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,23,18)(19,27,28,24)(20,26,22,25)(29,32,30,31)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,20)(18,26,21,25)(19,30,28,29)(24,32,27,31)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=ghi,f^2=h,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, 8663339);