Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,11,14,12,13,15,16)(17,18,20,24,21,22,19,23);(1,2)(3,7)(4,6)(5,8)(10,13)(11,15);(10,13)(14,16)(18,22)(23,24);(17,19,21,20)(18,23,22,24)Polycyclic presentation:
2^9|a^2=f,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=ci,e^a=eg,e^b=eh,f^b=fgh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67907);