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 | |
4579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,26,30,31)(27,33,35,29)(28,36,37,32)(34,39,40,38);(2,6)(4,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27,30,35)(26,29,31,33)(28,34,37,40)(32,39,36,38);(1,2)(3,4)(5,6)(7,8)(10,14)(11,18)(12,21)(22,23)(26,31)(28,37)(29,33)(34,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28)(26,32)(27,34)(29,39)(30,37)(31,36)(33,38)(35,40);(1,4)(2,3)(5,7)(6,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29)(26,27)(28,38)(30,33)(31,35)(32,40)(34,36)(37,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,29)(28,32)(30,31)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=gh,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, 8884321);