Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,12,16,21)(11,15,20,23,19,18,22,24);(2,5,7,6)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(9,12,13,21,14,10,16,17)(11,18,20,24,19,15,22,23);(10,12)(11,19)(17,21)(20,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,b^a=be,d^a=di,e^a=eh,g^a=gh,d^b=di,d^c=di140 conjugacy classes — view the full character table.
| maximal subgroup of | #72322064, #72322134, #72322139 |
| maximal normal subgroup of | #72322064, #72322134, #72322139 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90358);