Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,4)(5,6,7,8)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(9,10,12,14)(11,15,16,13)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(2,4)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287519, #72287653, #72287672, … |
| maximal normal subgroup of | #72287519, #72287653, #72287672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56643);