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 | |
4225 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)(9,10,15,14)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,26)(27,29)(28,32)(30,33)(31,35)(34,39)(36,37)(38,40);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,18)(23,24)(25,27,30,35)(26,29,33,31)(28,36,38,34)(32,37,40,39);(1,3,5,8)(2,6,4,7)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,38)(39,40);(1,4,5,2)(3,7,8,6)(9,12,15,21)(10,17,14,23)(11,20,22,13)(16,18,19,24)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,39,31,37);(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,17,19,23);(9,14,15,10)(11,19,22,16)(12,17,21,23)(13,24,20,18)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,36,40,34)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=hi,d^2=gh,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, 8374430);