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 | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61303 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(9,10,13,17,14,11,18,16)(12,19,20,23,21,15,22,24);(1,2,4,5)(3,7,8,6);(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,23)(22,24);(1,3,4,8)(2,6,5,7);(9,12,13,20,14,21,18,22)(10,15,17,24,11,19,16,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #41×#444, #72315597, #72315607, … |
| maximal normal subgroup of | #41×#444, #72315597, #72315607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291734);