Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,14,21,15,20,37,19)(11,22,27,39,26,30,25,16)(12,28,31,24,32,17,33,40)(13,34,35,23,36,29,38,18);(1,3,5,4)(2,6,7,8)(9,11,15,26)(10,16,20,39)(12,29,32,34)(13,28,36,17)(14,27,37,25)(18,33,23,31)(19,30,21,22)(24,38,40,35);(1,4,5,3)(2,6,7,8)(9,12,15,32)(10,17,20,28)(11,23,26,18)(13,30,36,22)(14,31,37,33)(16,38,39,35)(19,24,21,40)(25,34,27,29);(2,7)(6,8)(9,13,14,35,15,36,37,38)(10,18,19,29,20,23,21,34)(11,24,25,28,26,40,27,17)(12,30,31,16,32,22,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291865);