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 | |
4146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,13)(11,14)(15,16)(21,29)(25,30)(26,31)(27,32)(34,38)(35,41)(36,44)(46,48);(1,2,5,8)(3,6,10,9)(4,7,13,12)(11,14,16,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(3,10)(6,9)(11,16)(14,15)(18,23)(19,24)(25,30)(26,31)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019747);