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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4175 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,3,5,4)(2,6,7,8)(10,11)(15,17)(16,18)(22,23)(26,31)(27,33)(32,38)(34,39);(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(2,7)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,4,5,3)(2,8,7,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23);(1,2)(3,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=h,e^2=ghi,f^2=i,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, 8370387);