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 | |
4199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,27)(20,31)(21,26)(22,23)(24,30)(25,29)(28,32)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,19,23,30)(18,24,22,27)(20,28,29,21)(25,32,31,26)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(2,3)(4,11)(5,13)(14,15)(18,22)(20,29)(21,28)(24,27)(33,34,38,42)(35,39,41,37)(36,44,46,40)(43,45,48,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=i,e^2=h,f^2=g,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, 8901651);