Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6408564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #34364 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60647 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12,15,7)(4,16)(5,9)(6,10)(8,19)(11,18)(13,29,30,25)(14,24,21,23)(17,28,31,22)(20,26)(27,32);(1,3)(2,7)(4,17)(5,21)(6,15)(8,25)(9,24)(10,12)(11,23)(13,19)(14,18)(16,22)(20,31)(26,28)(27,29)(30,32);(1,4,18,32,6,20,5,19)(2,8,9,26,10,27,11,16)(3,13,21,31,15,30,14,17)(7,22,23,29,12,28,24,25);(2,9,10,11)(7,23,12,24)(8,26,27,16)(22,29,28,25);(1,3)(2,7)(4,13)(5,14)(6,15)(8,22)(9,23)(10,12)(11,24)(16,25)(17,19)(18,21)(20,30)(26,29)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,c^b=cg,d^b=df,g^b=gi,h^b=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315761);