Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,14)(4,11)(8,15,16,12)(10,13)(17,18,21,26,22,27,31,25)(19,23,20,29,28,24,30,32);(1,2,5,14,6,9,3,7)(4,12,13,8,11,15,10,16)(17,19,21,20,22,28,31,30)(18,23,26,29,27,24,25,32);(1,3,6,5)(2,7,9,14)(4,10,11,13)(8,12,16,15)(17,20,21,28,22,30,31,19)(18,24,25,29,27,23,26,32);(1,4,5,13,6,11,3,10)(2,8,7,12,9,16,14,15)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287474, #72287516, #72287521, … |
| maximal normal subgroup of | #72287474, #72287516, #72287521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138926);