Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
714 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)(3,4)(5,6,8,7)(9,10,14,19,17,23,24,12,16,20,13,18,15,11,21,22);(5,7,8,6)(9,11,16,23)(10,17,20,15)(12,21,22,24)(13,19,14,18);(2,4);(9,12,21,23,15,19,13,10,16,22,24,11,17,18,14,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72318753, #72319486, #72319645, … |
| maximal normal subgroup of | #72318753, #72319486, #72319645, … |
| maximal quotient of | #72322829 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405547);