Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,12,16,13,15,20,23)(11,14,18,21,17,22,19,24);(9,11)(10,14,15,22)(12,19)(13,17)(16,24,23,21)(18,20);(2,4)(5,6,7,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,f^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72280505, #72280546, #72280639, … |
| maximal normal subgroup of | #72280505, #72280546, #72280639, … |
| maximal quotient of | #72280786, #72303586 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30649);