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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60700 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,17,13,16,22,24)(11,18,20,21,19,15,23,12);(2,5)(3,8)(6,7)(9,11)(10,15)(12,17)(13,19)(14,20)(16,18)(21,24)(22,23);(9,12,14,18,13,21,22,15)(10,11,17,20,16,19,24,23);(11,19)(12,21)(15,18)(20,23);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=eh,f^a=fh,d^b=di,f^b=fh,d^c=di110 conjugacy classes — view the full character table.
| maximal subgroup of | #72306650, #72306653, #72306704, … |
| maximal normal subgroup of | #72306650, #72306653, #72306704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252336);