Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,14,20,15,21,13,19,16,22,30,38,32,12,18,31)(11,23,26,17,27,35,33,29,28,24,37,36,34,40,25,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,29)(13,26)(14,33)(15,34)(16,28)(18,37)(19,35)(20,24)(21,39)(22,36)(23,38)(25,30)(27,32)(31,40);(1,4,5,3)(2,6,7,8);(9,12,30,19,15,10,18,38,16,21,14,31,32,22,13,20)(11,24,26,36,27,40,33,39,28,23,37,17,34,35,25,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405241);