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 | |
3988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,14,16)(10,12,13,15)(34,37)(35,36)(41,45)(42,44);(1,3)(2,7)(4,9)(5,13)(6,11)(8,12)(10,14)(15,16)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,34)(35,40)(36,43)(37,39)(38,41)(42,47)(44,48)(45,46);(5,14)(8,16)(10,13)(12,15)(18,23)(20,29)(24,30)(26,32)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,16,13)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,5,6,14)(2,8,9,16)(3,10,11,13)(4,12,7,15)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,14)(15,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^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, 8525234);