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 | |
4349 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)(3,5)(4,8)(6,7)(9,10,14,15)(11,19,21,16)(12,17,22,20)(13,23,24,18)(26,27)(31,33)(32,34)(38,39);(2,7)(3,8)(10,15)(12,22)(16,19)(18,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3,6,8)(2,4,7,5)(10,15)(13,24)(16,19)(17,20)(29,37)(32,34)(35,40)(38,39);(1,4,6,5)(2,3,7,8)(9,11,14,21)(10,16,15,19)(12,23,22,18)(13,20,24,17)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,2)(3,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,5,6,4)(2,8,7,3)(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806048);