Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 16777216 |
| 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 | |
5626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,7)(10,11)(15,19)(18,20);(5,6,8,7)(9,10,12,11)(14,17)(18,20);(9,10,12,11)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,7,8,6)(9,11,12,10);(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72327811, #72327830, #72327867, … |
| maximal normal subgroup of | #72327811, #72327830, #72327867, … |
| maximal quotient of | #72331413, #72336217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215161);