Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C22×C42 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,11,12)(13,14,16,20,18,22,17,21)(15,19,24,27,23,26,25,28);(2,4)(13,15,17,25,18,23,16,24)(14,19,21,28,22,26,20,27);(5,6,7,8);(15,23)(19,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^b=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72321653, #72321690, #72321700, … |
| maximal normal subgroup of | #72321653, #72321690, #72321700, … |
| maximal quotient of | #72323083 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260864);