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 | |
4975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26)(27,33)(28,36)(29,34)(30,31)(32,37)(35,39)(38,40);(2,6)(3,7)(10,14)(15,17)(16,20)(23,24)(26,31)(28,37)(29,33)(35,38);(1,3)(2,4)(5,7)(6,8)(11,18)(15,17)(19,21)(23,24)(26,31)(27,34)(32,36)(35,38);(1,4)(2,3)(5,8)(6,7)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,27)(26,29)(28,38)(30,34)(31,33)(32,40)(35,37)(36,39);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,28)(26,32)(27,35)(29,39)(30,37)(31,36)(33,40)(34,38);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29)(26,27)(28,39)(30,33)(31,34)(32,35)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=i,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746709);