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 | |
4068 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,29)(23,25)(26,31)(27,32)(28,30)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(2,6)(7,9)(8,13)(14,15)(17,19,24,23)(18,25,21,20)(22,27,31,30)(26,32,29,28)(36,44)(40,46)(43,47)(45,48);(4,12)(8,13)(10,16)(14,15)(17,20)(18,23)(19,21)(22,28)(24,25)(26,30)(27,29)(31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,24,18)(19,25,23,20)(22,29,31,26)(27,32,30,28)(33,36,39,44)(34,40,38,46)(35,43,41,47)(37,45,42,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,23,24,19)(18,25,21,20)(22,30,31,27)(26,32,29,28)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8163793);