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 | |
3949 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)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34)(35,43)(36,41)(37,42)(38,39)(40,45)(44,46)(47,48);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(19,27)(21,30)(23,32)(25,31)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(2,7)(3,9)(8,15)(10,16)(17,19,22,27)(18,23,26,32)(20,21,28,30)(24,31,29,25)(33,34,39,38)(35,40,45,43)(36,46,44,41)(37,42,47,48);(1,2)(3,5)(4,8)(6,7)(9,14)(10,13)(11,16)(12,15)(17,20,22,28)(18,24,26,29)(19,21,27,30)(23,31,32,25)(33,36,39,44)(34,41,38,46)(35,37,45,47)(40,48,43,42);(1,4,6,12)(2,8,7,15)(3,10,9,16)(5,13,14,11)(17,21)(18,25)(19,28)(20,27)(22,30)(23,24)(26,31)(29,32)(33,37)(34,42)(35,44)(36,45)(38,48)(39,47)(40,41)(43,46);(1,5)(2,3)(4,11)(6,14)(7,9)(8,16)(10,15)(12,13)(17,18)(19,23)(20,24)(21,31)(22,26)(25,30)(27,32)(28,29)(33,38,39,34)(35,43,45,40)(36,46,44,41)(37,42,47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,d^2=hi,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, 9830834);