Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | |
5889 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)(6,8)(13,14,16,17)(15,19,20,18);(1,2)(3,4)(10,12)(14,17)(18,19);(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(5,6,7,8)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=g,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256744, #72256754, #72256765, … |
| maximal normal subgroup of | #72256744, #72256754, #72256765, … |
| maximal quotient of | #72323506, #72330005, #72330089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 781035);