Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,11,12,16,13,8,14,15)(17,18)(19,25,27,31)(20,23)(21,24)(22,26,29,32)(28,30);(2,7,10,9)(8,15,11,16)(17,19)(18,22,24,29)(20,26)(21,27)(23,25,30,31)(28,32);(1,3,6,13)(2,8,10,11)(4,12,5,14)(7,15,9,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,f^b=fi,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281134, #72281217, #72302143, … |
| maximal normal subgroup of | #72281134, #72281217, #72302143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12779);