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 | |
4251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,22,27)(18,23,21,25)(20,26,30,29)(24,31,28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(4,12)(7,14)(10,16)(13,15)(18,21)(23,25)(24,28)(31,32)(34,38)(39,41)(40,44)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202812);