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 | #41249 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,18)(11,16,12,13)(14,17,22,24)(19,23,20,21)(25,26)(27,33)(28,30)(29,36)(31,35)(32,38)(34,39)(37,40);(1,3,5,8)(2,6,7,4)(9,11,15,12)(10,13,18,16)(14,19,22,20)(17,21,24,23)(26,28)(31,33)(32,36)(39,40);(1,3)(2,4)(5,8)(6,7)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,4)(2,3)(5,6)(7,8)(9,12,15,11)(10,16,18,13)(14,19,22,20)(17,21,24,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(9,13,15,16)(10,12,18,11)(14,21,22,23)(17,20,24,19)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349732);