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 | #2540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4351 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)(12,20)(17,21)(18,24)(22,23)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38);(2,6)(3,7)(9,10,14,13)(11,16,19,15)(12,17,20,21)(18,23,24,22)(27,34)(28,36)(31,33)(32,37);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,18)(13,16)(14,19)(17,22)(20,24)(21,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,28,30,36)(26,32,29,37)(27,35,34,40)(31,38,33,39);(9,12,14,20)(10,17,13,21)(11,18,19,24)(15,22,16,23);(9,13,14,10)(11,16,19,15)(12,21,20,17)(18,23,24,22)(25,29,30,26)(27,33,34,31)(28,32,36,37)(35,38,40,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=gi,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, 8368735);