Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,24,22,25,30,31)(19,23,27,20,26,29,32,28);(2,5)(6,8)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32);(9,11,13,12)(10,14,15,16);(9,12,13,11)(10,14,15,16)(17,20,21,29,22,28,30,23)(18,19,24,27,25,26,31,32);(1,3,4,7)(2,6,5,8)(19,26)(20,28)(23,29)(27,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^b=ei,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281295);