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 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,19,21,16)(12,17,22,24)(13,20,23,18)(25,26,30,33)(27,34,29,31)(28,32,36,40)(35,39,37,38);(3,7)(6,8)(10,15)(12,22)(16,19)(18,20)(27,29)(31,34)(35,37)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,23)(16,19)(17,24)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,40,39);(9,11,14,21)(10,16,15,19)(12,18,22,20)(13,17,23,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,17,15,24)(11,20,21,18)(13,19,23,16)(25,29,30,27)(26,31,33,34)(28,35,36,37)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,19)(14,23)(15,20)(16,22)(21,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=g,e^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, 8805784);