Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,8)(9,10,12,11)(13,14,17,19)(15,20,16,18);(1,2)(3,4)(9,11,12,10)(13,15,17,16)(14,18,19,20);(2,4)(10,11)(13,14)(15,20)(16,18)(17,19);(6,8)(10,11)(14,19)(18,20);(13,16,17,15)(14,18,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280171, #72280271, #72280957, … |
| maximal normal subgroup of | #72280171, #72280271, #72280957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752065);