Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3572 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)(4,5)(7,8)(9,10)(11,16)(12,25)(13,19)(14,18)(15,20)(17,27)(21,33)(22,36)(23,31)(24,32)(26,39)(28,35)(29,34)(30,37)(38,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,33)(18,31)(19,36)(20,32)(21,27)(25,39)(28,38)(29,37)(30,34)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,28,15,29,13,27)(10,17,19,34,20,35,18,25)(11,21,22,37,24,38,23,26)(16,30,31,33,32,39,36,40);(3,7)(6,8)Polycyclic presentation:
2^9|c^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289574);