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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3819 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,27)(19,26,22,21)(20,25,31,24)(28,29,30,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19,23,22)(18,21,27,26)(20,28,31,30)(24,29,25,32)(34,39)(36,40)(41,45)(44,47);(2,4)(3,9)(8,12)(11,13)(17,20,23,31)(18,24,27,25)(19,28,22,30)(21,29,26,32)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,4,7,2)(3,10,9,6)(5,8,15,12)(11,16,13,14)(17,20)(18,25)(19,28)(21,32)(22,30)(23,31)(24,27)(26,29)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,5)(2,8)(3,11)(4,12)(6,16)(7,15)(9,13)(10,14)(17,21)(18,19)(20,29)(22,27)(23,26)(24,28)(25,30)(31,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,22,23,19)(18,26,27,21)(20,30,31,28)(24,32,25,29)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113812);