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 | |
4483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,17,20,15)(12,21,23,13)(18,24,22,19)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,16,17)(12,22,23,18)(13,19,21,24)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(2,3)(6,7)(10,16)(12,23)(15,17)(18,22)(28,37)(32,40)(35,36)(38,39);(4,8)(6,7)(9,11)(10,15)(12,18)(13,24)(14,20)(16,17)(19,21)(22,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,18)(14,23)(15,24)(16,21)(17,19)(20,22)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,19)(14,21)(15,22)(16,23)(17,18)(20,24)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=ghi,f^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, 8521407);