Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | #2358 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1256 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)(3,7)(4,6)(5,8)(9,10)(11,23)(12,27,29,36)(13,31,32,38)(14,20)(15,21)(16,22)(17,25)(18,24,40,30)(19,28,35,33)(26,39)(34,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,28)(13,30)(14,25)(15,26)(16,34)(18,38)(19,27)(20,23)(21,37)(22,39)(24,32)(29,33)(31,40)(35,36);(9,12,15,30,14,29,16,24)(10,18,21,36,20,40,22,27)(11,13,26,33,25,32,34,28)(17,35,37,31,23,19,39,38);(2,6)(3,8)(9,13)(10,19)(11,24)(12,26)(14,32)(15,33)(16,28)(17,36)(18,39)(20,35)(21,38)(22,31)(23,27)(25,30)(29,34)(37,40)Polycyclic presentation:
2^9|a^2=h,c^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296181);