Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(6,8)(14,17)(15,19);(2,3)(10,12)(13,14,16,17)(15,20,19,18);(1,2,4,3)(9,10,11,12)(13,14)(15,20)(16,17)(18,19);(5,6)(7,8)(13,14)(15,20)(16,17)(18,19);(1,3,4,2)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|b^2=f,c^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300643, #72300667, #72330816, … |
| maximal normal subgroup of | #72300643, #72300667, #72330816, … |
| maximal quotient of | #72332779 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3944316);