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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3713 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,18,23,21)(19,26,25,22)(20,28,31,24)(27,30,29,32)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(3,10)(6,7)(11,16)(14,15)(17,19,23,25)(18,22,21,26)(20,29,31,27)(24,32,28,30)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18,23,21)(19,22,25,26)(20,28,31,24)(27,32,29,30)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,31)(18,24,21,28)(19,27,25,29)(22,30,26,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,23,18)(19,22,25,26)(20,24,31,28)(27,32,29,30)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22)(18,25)(19,21)(20,30)(23,26)(24,29)(27,28)(31,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227601);