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 | |
3884 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(33,34,39,41)(35,40,45,48)(36,42,43,37)(38,44,47,46);(2,4)(3,10)(8,12)(11,13)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35,39,45)(34,40,41,48)(36,44,43,46)(37,38,42,47);(2,4)(7,9)(8,12)(15,16)(18,23)(20,29)(24,31)(26,32)(34,41)(35,45)(37,42)(44,46);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36,39,43)(34,42,41,37)(35,46,45,44)(38,48,47,40);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,44,48,46);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,40,43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540533);