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, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(1,3,4,2)(6,7);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(5,7,8,6)(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=de,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72318753, #72319542, #72319586, … |
| maximal normal subgroup of | #72318753, #72319542, #72319586, … |
| maximal quotient of | #72322811 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405518);