Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,16,13,21)(19,24,22,23)(25,26)(27,34)(28,32)(29,37)(30,33)(31,35)(36,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24);(3,7)(5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(3,7)(5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34);(2,6)(5,8)(10,17)(15,18)(16,21)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6540558);