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 | |
3910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,12,14,8)(11,15,13,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34)(35,36)(37,42)(38,43)(39,40)(41,48)(44,46)(45,47);(2,5)(7,9)(8,12)(15,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(34,40)(35,41)(42,43)(44,45);(3,10)(7,9)(11,13)(15,16)(18,19)(23,25)(24,26)(30,31)(33,35,39,41)(34,36,40,48)(37,44,38,45)(42,46,43,47);(1,3)(2,7)(4,13)(5,9)(6,10)(8,16)(11,14)(12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,41)(35,40)(37,46)(38,47)(39,48)(42,45)(43,44);(1,4,6,14)(2,8,5,12)(3,11,10,13)(7,15,9,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37,39,38)(34,42,40,43)(35,44,41,45)(36,46,48,47);(1,5,6,2)(3,9,10,7)(4,12,14,8)(11,16,13,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,37)(34,43,40,42)(35,45,41,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297619);