Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,8)(9,10,11,12)(14,17)(15,19);(2,4)(6,8)(9,10)(11,12)(14,17)(18,20);(1,2)(3,4)(5,6,7,8);(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=f,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72316142, #72317076, #72323279, … |
| maximal normal subgroup of | #72316142, #72317076, #72323279, … |
| maximal quotient of | #72317079, #72323533, #72323535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5862525);