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:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,23,20)(19,24,26,21)(22,28,31,25)(27,29,32,30)(34,38)(36,44)(39,41)(42,45);(4,11)(8,14)(10,15)(12,16)(17,19,23,26)(18,24,20,21)(22,27,31,32)(25,30,28,29)(33,34,37,38)(35,41,43,39)(36,40,44,46)(42,48,45,47);(2,7)(3,9)(8,14)(10,15)(18,20)(19,26)(25,28)(27,32)(33,34,37,38)(35,39,43,41)(36,40,44,46)(42,47,45,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,18)(19,24,26,21)(22,28,31,25)(27,30,32,29)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,24)(18,19,20,26)(22,30,31,29)(25,32,28,27)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,44)(46,47);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,36,37,44)(34,40,38,46)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154035);