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 | |
3779 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29)(34,39)(35,41)(37,45)(44,46);(1,3,7,4)(2,8,9,10)(5,12,15,13)(6,11,16,14)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(33,34,38,39)(35,40,41,36)(37,42,45,47)(43,46,48,44);(2,9)(3,4)(11,14)(12,13)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,14,16,11)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,47)(45,48);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=i,e^2=h,f^2=gh,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, 10050469);