Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(6,7)(10,11);(2,4)(14,16);(5,6,8,7)(9,10,12,11)(13,14)(15,16);(2,4)(9,10,12,11);(1,2)(3,4)(5,7,8,6)(9,11,12,10)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232137, #72232166, #72232180, … |
| maximal normal subgroup of | #72232137, #72232166, #72232180, … |
| maximal quotient of | #72278012, #72278149, #72278228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1401306);