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 | #45094 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,7)(3,4)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38);(1,2,6,7)(3,5,4,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(27,34)(28,36)(31,33)(32,37);(1,3,6,4)(2,5,7,8)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,4,6,3)(2,5,7,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,28,30,36)(26,32,29,37)(27,35,34,40)(31,38,33,39);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(1,5)(2,4)(3,7)(6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,26)(27,33,34,31)(28,32,36,37)(35,38,40,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8362934);