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 | |
4036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(17,18,22,28)(19,26,30,23)(20,25,31,24)(21,29,32,27)(35,41)(38,45)(42,47)(46,48);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,28)(27,31)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(2,7)(3,9)(8,15)(10,16)(17,20,22,31)(18,24,28,25)(19,29,30,27)(21,23,32,26)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,2)(3,5)(4,8)(6,7)(9,14)(10,13)(11,16)(12,15)(17,20)(18,25)(19,29)(21,23)(22,31)(24,28)(26,32)(27,30)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4,6,12)(2,8,7,15)(3,10,9,16)(5,13,14,11)(17,19)(18,26)(20,29)(21,24)(22,30)(23,28)(25,32)(27,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,5)(2,3)(4,11)(6,14)(7,9)(8,16)(10,15)(12,13)(17,21,22,32)(18,27,28,29)(19,25,30,24)(20,23,31,26)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820797);