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) = C29 |
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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,23)(22,24);(1,3,4,2)(6,7);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(5,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,24,19);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,19,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fi,e^2=i,g^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 | C4×#61066, #72284735, #72284895 |
| maximal normal subgroup of | C4×#61066, #72284735, #72284895 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289053);