Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,13,18,15,19,14,29)(11,20,23,37,22,36,30,34)(12,24,27,35,26,39,28,17)(16,25,32,38,31,40,33,21);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,25)(13,23)(14,30)(15,22)(17,34)(18,32)(19,31)(20,24)(21,28)(26,40)(27,38)(29,33)(35,37)(36,39);(1,3,4,8)(2,5,6,7)(12,26)(17,35)(20,36)(21,38)(24,39)(25,40)(27,28)(34,37)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56652);