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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,12)(5,8,14,16)(10,11,13,15)(17,18)(19,26)(20,30)(21,25)(22,23)(24,29)(27,28)(31,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3,6,7)(2,4,9,12)(5,13,14,10)(8,15,16,11)(18,22)(21,30)(24,26)(28,32)(34,39)(35,41)(40,46)(42,47);(1,3)(2,4)(5,10)(6,7)(8,11)(9,12)(13,14)(15,16)(17,19)(18,24)(20,31)(21,32)(22,26)(23,29)(25,27)(28,30)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(17,20)(18,21)(19,27)(22,30)(23,25)(24,28)(26,32)(29,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,13)(10,16)(14,15)(17,21,23,30)(18,25,22,20)(19,28,29,32)(24,31,26,27);(1,5)(2,8)(3,10)(4,11)(6,14)(7,13)(9,16)(12,15)(17,22,23,18)(19,26,29,24)(20,30,25,21)(27,32,31,28)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150872);