Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(6,8)(13,14,18,19)(15,17,16,20);(10,12)(14,19)(17,20);(1,2,4,3)(9,10)(11,12)(13,15,18,16)(14,17,19,20);(1,2,4,3)(5,6)(7,8)(13,16,18,15)(14,17,19,20);(1,3,4,2)(13,17)(14,16)(15,19)(18,20)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72282285, #72282308, #72298036, … |
| maximal normal subgroup of | #72282285, #72282308, #72298036, … |
| maximal quotient of | #72298270, #72298285 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1326425);