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 | |
4604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,21,16)(12,13,17,22)(19,24,23,20)(25,26)(27,32)(28,30)(29,34)(31,33)(35,37)(36,39)(38,40);(2,5)(6,7)(10,15)(12,17)(16,18)(19,23)(25,27,31,28)(26,32,33,30)(29,35,38,36)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(26,33)(27,28)(34,40)(35,36);(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,39,40,37);(1,2)(3,7)(4,5)(6,8)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,24)(21,23)(25,29)(26,34)(27,35)(28,36)(30,39)(31,38)(32,37)(33,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,30)(26,27)(28,33)(29,37)(31,32)(34,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9686324);