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 | |
4660 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)(11,19)(12,17)(13,20)(14,15)(16,22)(18,24)(21,23)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,23)(16,19)(18,20)(25,27)(26,33)(28,38)(29,35)(30,39)(31,34)(32,40)(36,37);(2,5)(6,8)(10,15)(13,24)(16,19)(17,21)(25,28,32,29)(26,34,36,30)(27,38,40,35)(31,37,39,33);(3,7)(6,8)(9,11,14,22)(10,16,15,19)(12,18,23,20)(13,21,24,17)(25,29,32,28)(26,34,36,30)(27,35,40,38)(31,37,39,33);(1,3,4,7)(2,6,5,8)(9,12,14,23)(10,17,15,21)(11,20,22,18)(13,16,24,19)(25,30,32,34)(26,29,36,28)(27,39,40,31)(33,35,37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,15,20)(11,21,22,17)(12,16,23,19)(25,31,32,39)(26,35,36,38)(27,34,40,30)(28,33,29,37)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=gi,f^2=g,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, 8806246);