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 | |
4070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,5,10)(4,9,12,15)(11,14,13,16)(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,28,32,26);(1,3)(2,7)(4,11)(5,6)(8,10)(9,14)(12,13)(15,16)(18,23)(20,31)(24,27)(26,28)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(2,8)(3,5)(4,12)(14,16)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,25,32,29)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,4,6,12)(2,9,8,15)(3,11,5,13)(7,14,10,16)(17,19)(18,24)(20,26)(21,29)(22,30)(23,27)(25,32)(28,31)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,5,6,3)(2,10,8,7)(4,11,12,13)(9,14,15,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,30)(27,31)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=i,e^2=gh,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, 8901758);