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 | |
4039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,11,12,7)(4,8,14,16)(5,13,15,9)(18,20)(21,27)(25,28)(29,32)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(3,12)(5,15)(7,11)(9,13)(17,18,23,20)(19,21,24,27)(22,28,30,25)(26,32,31,29)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,3)(2,7)(4,13)(5,8)(6,12)(9,14)(10,11)(15,16)(17,19,23,24)(18,21,20,27)(22,26,30,31)(25,29,28,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,4)(2,8)(3,9)(5,11)(6,14)(7,15)(10,16)(12,13)(17,20,23,18)(19,21,24,27)(22,28,30,25)(26,29,31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,3)(2,7)(4,13)(5,8)(6,12)(9,14)(10,11)(15,16)(17,21)(18,24)(19,20)(22,29)(23,27)(25,31)(26,28)(30,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48);(1,5)(2,9)(3,8)(4,11)(6,15)(7,14)(10,13)(12,16)(17,22,23,30)(18,25,20,28)(19,26,24,31)(21,29,27,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=ghi,f^2=g,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=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757883);