Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×AGL(1,ℤ/8) | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,8,5,9,4,12)(3,7,10,16,11,15,13,14)(17,18,22,26,23,20,31,27)(19,21,29,28,30,32,25,24);(1,2,4,12,5,9,6,8)(3,7,13,14,11,15,10,16)(17,19,23,30)(18,24,20,28)(21,26,32,27)(22,25,31,29);(1,3)(2,7)(4,13)(5,11)(6,10)(8,16)(9,15)(12,14)(17,20)(18,23)(19,28)(21,25)(22,27)(24,30)(26,31)(29,32);(1,4,5,6)(2,8,9,12)(3,10,11,13)(7,14,15,16)(17,21,31,24,23,32,22,28)(18,25,27,30,20,29,26,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 137259);