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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(18,23)(19,24)(26,31)(27,33)(29,36)(34,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,26,30,31)(27,32,33,38)(28,36,37,29)(34,40,39,35);(1,3,5,6)(2,4,7,8)(25,27,30,33)(26,32,31,38)(28,34,37,39)(29,35,36,40);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,27)(26,32)(28,34)(29,35)(30,33)(31,38)(36,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,28,30,37)(26,29,31,36)(27,34,33,39)(32,35,38,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,29)(26,28)(27,35)(30,36)(31,37)(32,34)(33,40)(38,39)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9590966);