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 | |
4231 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)(8,13)(10,16)(12,14)(17,18)(19,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,34,40,39)(35,43,37,41)(36,42,46,48)(38,44,45,47);(2,7)(4,14)(8,11)(10,15)(17,19,23,31)(18,24,22,30)(20,28,29,21)(25,27,26,32)(33,35,40,37)(34,41,39,43)(36,45,46,38)(42,44,48,47);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,9,16,12)(17,20,23,29)(18,25,22,26)(19,21,31,28)(24,32,30,27)(33,36,40,46)(34,42,39,48)(35,45,37,38)(41,44,43,47);(1,3)(2,8)(4,15)(5,12)(6,13)(7,11)(9,16)(10,14)(17,20)(18,26)(19,28)(21,31)(22,25)(23,29)(24,32)(27,30)(33,37,40,35)(34,43,39,41)(36,45,46,38)(42,44,48,47);(1,4,6,14)(2,9,7,12)(3,10,13,15)(5,11,16,8)(17,21,23,28)(18,27,22,32)(19,29,31,20)(24,25,30,26)(33,38,40,45)(34,44,39,47)(35,36,37,46)(41,48,43,42);(1,5)(2,10)(3,12)(4,11)(6,16)(7,15)(8,14)(9,13)(17,22,23,18)(19,30,31,24)(20,25,29,26)(21,32,28,27)(33,39,40,34)(35,43,37,41)(36,42,46,48)(38,47,45,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=ghi,e^2=gi,f^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, 8922591);