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 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,12)(13,22,21,18)(14,19,23,24)(25,26)(27,35)(28,38)(29,40)(30,34)(31,37)(32,39)(33,36);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,17)(13,23)(14,21)(15,20)(18,19)(22,24)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(2,5)(6,8)(10,17)(11,20)(13,21)(19,24)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,33,35,40);(3,7)(6,8)(9,12)(10,11)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,29)(26,33)(27,28)(30,36)(31,38)(32,35)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,21)(10,18,17,22)(11,14,20,23)(12,19,16,24)(25,28)(26,32)(27,29)(30,39)(31,40)(33,35)(34,38)(36,37);(1,3)(2,6)(4,7)(5,8)(9,14)(10,19)(11,21)(12,22)(13,20)(15,23)(16,18)(17,24)(25,26)(27,31)(28,32)(29,40)(30,34)(33,36)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8499430);