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 | |
4058 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)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34,39,42)(35,43,45,40)(36,46,47,37)(38,41,48,44);(1,3,7,11)(2,8,6,14)(4,12,13,5)(9,16,15,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,40)(36,44)(37,38)(39,45)(41,47)(42,43)(46,48);(2,6)(3,11)(4,13)(10,16)(19,25)(20,28)(23,27)(24,29)(33,34,39,42)(35,40,45,43)(36,46,47,37)(38,44,48,41);(1,4)(2,9)(3,12)(5,11)(6,15)(7,13)(8,16)(10,14)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,36,39,47)(34,37,42,46)(35,44,45,41)(38,43,48,40);(1,5,7,12)(2,10,6,16)(3,13,11,4)(8,15,14,9)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37)(34,36)(35,38)(39,46)(40,44)(41,43)(42,47)(45,48);(1,6,7,2)(3,14,11,8)(4,15,13,9)(5,16,12,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38)(34,41)(35,37)(36,43)(39,48)(40,47)(42,44)(45,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8830124);