Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60559 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,11)(3,12,4,9)(5,10,6,15)(8,14,16,13)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(19,25)(22,28)(26,31)(29,32);(2,11)(8,16)(9,12)(10,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dhi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252145);