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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,33)(27,34,36,39)(28,32,29,37)(31,38,40,35);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(3,7)(5,8)(10,16)(11,19)(15,22)(18,24);(2,6)(5,8)(10,16)(14,17)(15,22)(21,23)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,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, 6539685);