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 | |
3889 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)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,28,31)(20,25,30,29)(21,22,26,32)(33,34)(35,46)(36,39)(37,47)(38,43)(40,45)(41,44)(42,48);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19)(18,24)(20,22)(21,29)(23,28)(25,26)(27,31)(30,32)(33,35)(34,38)(36,47)(37,41)(39,48)(40,43)(42,44)(45,46);(2,8)(3,11)(4,13)(10,14)(19,28)(20,30)(24,31)(25,29)(33,36,40,44)(34,41,45,39)(35,47,43,42)(37,46,48,38);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20)(18,25)(19,22)(21,31)(23,30)(24,26)(27,29)(28,32)(33,37,40,48)(34,42,45,47)(35,41,43,39)(36,38,44,46);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21)(18,22)(19,29)(20,24)(23,26)(25,28)(27,32)(30,31)(33,38)(34,43)(35,45)(36,37)(39,47)(40,46)(41,42)(44,48);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,22)(18,26)(19,20)(21,27)(23,32)(24,25)(28,30)(29,31)(33,39,40,41)(34,44,45,36)(35,48,43,37)(38,42,46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034797);