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 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34,39,41)(35,40,45,48)(36,42,43,37)(38,44,47,46);(2,6)(3,10)(8,14)(11,12)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,35,39,45)(34,40,41,48)(36,44,43,46)(37,38,42,47);(2,6)(7,9)(8,14)(15,16)(21,30)(24,32)(26,29)(28,31)(34,41)(35,45)(37,42)(44,46);(1,3)(2,7)(4,12)(5,10)(6,9)(8,16)(11,13)(14,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,43)(34,42,41,37)(35,46,45,44)(38,48,47,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,44,48,46);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,40,43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529428);