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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,27)(19,23,29,31)(20,25,30,24)(21,28,32,26)(33,34)(35,40)(36,43)(37,39)(38,45)(41,47)(42,44)(46,48);(1,2,6,7)(3,8,10,15)(4,5,9,14)(11,16,13,12)(17,19,22,29)(18,23,27,31)(20,26,30,28)(21,24,32,25)(33,35,39,43)(34,36,37,40)(38,42,47,48)(41,44,45,46);(2,7)(3,10)(5,14)(11,13)(17,20,22,30)(18,24,27,25)(19,28,29,26)(21,31,32,23)(33,36,39,40)(34,35,37,43)(38,46,47,44)(41,48,45,42);(1,3)(2,8)(4,13)(5,12)(6,10)(7,15)(9,11)(14,16)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,46,48,44);(1,4,6,9)(2,5,7,14)(3,11,10,13)(8,16,15,12)(17,19)(18,23)(20,28)(21,24)(22,29)(25,32)(26,30)(27,31)(33,35)(34,40)(36,37)(38,42)(39,43)(41,46)(44,45)(47,48);(1,5)(2,9)(3,12)(4,7)(6,14)(8,11)(10,16)(13,15)(17,21,22,32)(18,26,27,28)(19,24,29,25)(20,23,30,31)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826518);