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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,4,8)(5,6,15,16)(11,14,13,12)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,38,41)(35,39,36,43)(37,40,46,45)(42,47,44,48);(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3,7,4)(2,8,9,10)(5,11,15,13)(6,12,16,14)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(1,4,7,3)(2,8,9,10)(5,11,15,13)(6,14,16,12)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,34)(35,39)(36,43)(37,45)(38,41)(40,46)(42,48)(44,47);(1,5)(2,6)(3,11)(4,13)(7,15)(8,12)(9,16)(10,14)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,6)(2,5)(3,12)(4,14)(7,16)(8,11)(9,15)(10,13)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327274);