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 | #35144 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(29,37)(34,38)(35,36)(39,40);(2,6)(3,7)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(26,31)(32,33)(34,38)(39,40);(1,3,5,7)(2,4,6,8)(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,39)(37,38);(1,4)(2,7)(3,6)(5,8)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038427);