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 | #44260 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4051 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)(3,5,10,8)(4,9,14,16)(11,15,12,13)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(34,39)(35,42)(37,45)(44,47);(2,7)(3,10)(9,16)(11,12)(20,28)(24,31)(26,32)(29,30)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154183);