Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2292 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)(5,6,8,9)(7,10,11,12);(6,9)(10,12)(13,14,18,19)(15,16,20,17);(2,4)(14,15)(16,17)(19,20);(5,7,8,11)(6,10,9,12)(13,15)(14,16)(17,19)(18,20);(13,16)(14,19)(17,18)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72305240, #72305250, #72305274, … |
| maximal normal subgroup of | #72305240, #72305250, #72305274, … |
| maximal quotient of | #72305603, #72305716 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483342);