Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| 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 | |
561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,5,9,4,8)(3,7,11,14,12,15,13,16)(17,18,21,24,22,19,26,25)(20,27,29,32,30,23,31,28);(1,2,4,8,5,9,6,10)(3,7,13,16,12,15,11,14)(17,19)(18,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32);(1,3)(2,7)(4,13)(5,12)(6,11)(8,16)(9,15)(10,14)(17,18)(19,22)(20,27)(21,24)(23,30)(25,26)(28,31)(29,32);(1,4,5,6)(2,8,9,10)(3,11,12,13)(7,14,15,16)(17,20,21,29,22,30,26,31)(18,23,24,28,19,27,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dei,e^b=eh,g^b=gh,d^c=di,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 137280);