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 | |
4194 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(33,34,37,41)(35,39,43,38)(36,40,45,44)(42,47,48,46);(2,7)(4,14)(8,11)(10,15)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,7)(3,12)(4,14)(5,16)(18,23)(19,27)(20,30)(21,31)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8884501);