Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2582 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)(7,10)(8,9)(11,12)(13,14,18,20,17,16,15,19);(1,2,3,4)(6,9)(7,11)(14,16)(19,20);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,19,16,20);(6,9)(10,12);(13,16,18,19,17,14,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,e^2=g,g^2=i,b^a=bf,d^a=dfi,c^b=cfi,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72313088, #72313104, #72313112, … |
| maximal normal subgroup of | #72313088, #72313104, #72313112, … |
| maximal quotient of | #72313298, #72313670 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277705);