Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,12,15)(11,14,20,22)(13,21,19,17)(16,24,23,18);(10,14)(11,18)(13,19)(15,22)(16,17)(20,24)(21,23);(1,3,4,8,5,2,6,7)(10,15)(14,22)(16,23)(17,21);(2,3)(7,8)Polycyclic presentation:
2^9|a^2=f,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232027, #72234323, #72234324, … |
| maximal normal subgroup of | #72232027, #72234323, #72234324, … |
| maximal quotient of | #72234323, #72234332, #72234343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391269);