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 |
| prime radical | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,21)(14,17)(15,16)(18,24)(19,22)(20,23)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(13,22)(20,24)(26,33)(28,40)(32,36)(35,37);(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7720349);