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 | #39747 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,21)(12,18)(13,15)(14,23)(16,20)(17,24)(19,22)(25,26,30,31)(27,28,35,32)(29,33,38,39)(34,36,40,37);(1,2,5,4)(3,7,8,6)(9,11)(10,17)(12,14)(13,22)(15,23)(16,24)(18,19)(20,21)(26,31)(28,32)(33,39)(36,37);(1,3,5,8)(2,6,4,7)(9,12,16,13)(10,18,20,15)(11,14,24,22)(17,19,21,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13,16,12)(10,18,20,15)(11,22,24,14)(17,19,21,23)(25,28,30,32)(26,27,31,35)(29,37,38,36)(33,40,39,34);(9,14,16,22)(10,19,20,23)(11,12,24,13)(15,17,18,21);(9,15,16,18)(10,13,20,12)(11,23,24,19)(14,17,22,21)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=g,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, 8368525);