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 | #43966 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,39,44)(35,40,37,42)(36,46,45,41)(38,43,47,48);(2,7)(4,9)(8,13)(12,14)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,43,46,48);(2,7)(3,10)(8,13)(11,16)(18,24)(20,25)(23,30)(26,31)(33,36,39,45)(34,41,44,46)(35,38,37,47)(40,48,42,43);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,37,39,35)(34,42,44,40)(36,38,45,47)(41,48,46,43);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,38)(34,43)(35,45)(36,37)(39,47)(40,41)(42,46)(44,48);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,34)(35,40)(36,41)(37,42)(38,48)(39,44)(43,47)(45,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013890);