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 | |
4289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,31,29)(33,34,38,36)(35,40,43,39)(37,41,46,44)(42,48,45,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35)(34,39)(36,40)(37,42)(38,43)(41,47)(44,48)(45,46);(2,9)(3,4)(10,16)(11,12)(19,23)(20,26)(27,31)(28,29)(33,34,38,36)(35,39,43,40)(37,41,46,44)(42,47,45,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,36,38,34)(35,39,43,40)(37,44,46,41)(42,47,45,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,35)(34,40)(36,39)(37,45)(38,43)(41,47)(42,46)(44,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962942);