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 | |
4693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,8)(10,11)(13,14)(15,16)(17,19)(18,20);(1,2,3,4)(13,15,17,20)(14,18,19,16);(9,10,12,11)(13,14)(15,16)(17,19)(18,20);(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,14)(15,18)(16,20)(17,19);(1,2)(3,4)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,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 | #72280272, #72280331, #72280449, … |
| maximal normal subgroup of | #72280272, #72280331, #72280449, … |
| maximal quotient of | #72299533, #72299621, #72299694, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 731958);