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 | |
4226 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(34,39)(36,42)(41,45)(44,46);(2,3)(7,9)(8,10)(14,15)(18,19)(23,25)(24,26)(31,32)(33,35,38,40)(34,36,39,42)(37,43,47,48)(41,44,45,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,36)(34,40)(35,39)(37,46)(38,42)(41,43)(44,47)(45,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37,38,47)(34,41,39,45)(35,43,40,48)(36,44,42,46)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10088501);