Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
5849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,8)(10,11)(13,14,17,16)(15,18,20,19);(2,4)(13,14)(15,19)(16,17)(18,20);(9,10,12,11)(13,15)(14,18)(16,19)(17,20);(2,4)(5,6,7,8)(9,11,12,10)(13,15)(14,19)(16,18)(17,20);(1,2)(3,4)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233787, #72234010, #72234011, … |
| maximal normal subgroup of | #72233787, #72234010, #72234011, … |
| maximal quotient of | #72278166, #72298230, #72298289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 732170);