Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,13,15)(11,16,19,23)(12,14,20,17)(18,24,21,22)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,3,5,4)(2,6,7,8)(10,14)(11,18)(12,20)(15,17)(16,23)(19,21)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,4,5,3)(2,6,7,8)(10,15)(14,17)(16,23)(22,24)(27,33)(30,36)(34,39)(37,40);(9,11,13,19)(10,16,15,23)(12,21,20,18)(14,22,17,24)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=de,e^b=eh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 133849);