Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128001 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,19)(3,11)(4,14)(8,20,21,32)(10,15,23,28)(12,26)(13,25)(16,29)(18,30,31,22)(24,27);(1,2)(3,8)(4,15,16,7)(5,9)(6,10)(11,22)(12,20,24,18)(13,21)(14,28,29,19)(17,23)(25,30)(26,31,27,32);(1,3,5,13)(2,8,9,21)(4,12,14,27)(6,11,17,25)(7,18,19,31)(10,22,23,30)(15,20,28,32)(16,24,29,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,d^b=dh,e^b=eg,h^b=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72236576, #72236736, #72238659, … |
| maximal normal subgroup of | #72236576, #72236736, #72238659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35240);