Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,9,11)(7,12,8,10);(5,7,9,8)(6,10,11,12)(13,14)(15,17)(16,18)(19,20);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(15,20)(18,19);(14,17)(15,20)(18,19);(14,18)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72250261, #72256996, #72257029, … |
| maximal normal subgroup of | #72250261, #72256996, #72257029, … |
| maximal quotient of | #72305855, #72306062, #72308531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346281);