Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,36)(27,37,38,33)(28,34,29,39)(30,35,32,40);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27)(26,33)(28,32)(29,30)(31,38)(34,35)(36,37)(39,40);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,29)(26,33,36,37)(27,30,38,32)(34,35,39,40);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(25,29,31,28)(26,34,36,39)(27,30,38,32)(33,35,37,40);(25,30,31,32)(26,35,36,40)(27,28,38,29)(33,39,37,34)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342014);