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 | |
4620 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(1,2,4,6)(3,5,8,7)(10,15)(12,17)(16,21)(20,23)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(13,22)(20,23)(26,31)(30,37)(32,35)(38,39);(1,3,4,8)(2,5,6,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,28)(26,32)(27,33)(29,36)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9343621);