Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(9,10)(11,21,25,32)(12,17)(13,19)(14,20)(15,18)(16,23,31,24)(22,37,39,33)(26,30,38,40)(27,35)(28,36)(29,34);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,26)(13,24)(14,23)(15,25)(17,33)(18,31)(19,32)(20,21)(22,28)(27,39)(29,38)(30,36)(34,37)(35,40);(1,3,4,2)(6,8)(9,12,13,27,15,29,14,28)(10,17,19,35,18,34,20,36)(11,22,23,38,25,39,24,26)(16,30,21,37,31,40,32,33);(10,18)(16,31)(17,34)(19,20)(21,32)(30,40)(33,37)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265506);