Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636816 |
| 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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1474 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,29,32)(27,33,35,40)(28,37,34,31)(30,38,39,36);(1,3)(2,5)(4,8)(6,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23);(3,8)(5,7)(25,27,34,38,29,35,28,36)(26,30,37,40,32,39,31,33);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(27,35)(30,39)(33,40)(36,38);(10,16)(14,17)(15,20)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gi,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6556289);