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 | #55405 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,5)(7,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,34)(27,31,36,37)(28,32,38,40)(29,33,39,35);(1,2,6,4)(3,8,5,7)(10,12)(15,17)(16,20)(22,23)(25,27,30,36)(26,31,34,37)(28,33,38,35)(29,32,39,40);(1,3,6,5)(2,7,4,8)(25,27)(26,31)(28,35)(29,32)(30,36)(33,38)(34,37)(39,40);(1,2,6,4)(3,8,5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(1,4,6,2)(3,8,5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28)(26,32)(27,35)(29,31)(30,38)(33,36)(34,40)(37,39);(1,5,6,3)(2,7,4,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29)(26,33)(27,32)(28,37)(30,39)(31,38)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,e^2=gh,f^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, 9701431);