Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2294 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)(7,9)(8,10)(11,12)(13,14,17,15,18,19,16,20);(5,7,8,11)(6,9,10,12)(14,15)(16,17)(19,20);(6,10)(9,12);(13,15,16,14,18,20,17,19);(14,19)(15,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #117×AGL(1,ℤ/8), #72305412, #72305419 |
| maximal normal subgroup of | #117×AGL(1,ℤ/8), #72305412, #72305419 |
| maximal quotient of | #72306190, #72306195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287065);