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 | #41256 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,13)(12,17)(14,16)(15,19)(18,21)(20,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,22)(21,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,5,4)(2,6,8,7)(10,16)(13,15)(17,24)(21,23)(26,31)(29,38)(32,39)(36,40);(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,18)(13,21)(14,22)(15,23)(16,24)(19,20)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,13,14,15)(10,11,16,19)(12,21,22,23)(17,18,24,20)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gh,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7559978);