Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#121 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9,11,12)(13,14,16,20,17,21,24,27,18,15,19,22,25,23,26,28);(5,6,10,9,8,11,7,12)(13,15,16,22,17,23,24,28,18,14,19,20,25,21,26,27);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^b=dg,f^b=fg152 conjugacy classes — view the full character table.
| maximal subgroup of | #72287273, #72323186 |
| maximal normal subgroup of | #72287273, #72323186 |
| maximal quotient of | #72287158 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43785);