Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437793 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,9,15,10,14,11,12)(7,16,17,19,8,13,18,20);(6,12)(7,17)(8,18)(9,11)(14,15)(19,20);(2,4)(5,7)(6,13)(8,10)(9,17)(11,18)(12,19)(14,16)(15,20);(5,8)(6,13)(7,10)(9,18)(11,17)(12,19)(14,16)(15,20);(6,14)(12,15)(13,16)(19,20)Polycyclic presentation:
2^9|a^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72213751, #72213852, … |
| maximal normal subgroup of | #72213527, #72213751, #72213852, … |
| maximal quotient of | #72238755, C2×#6437793, #72248325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344989);