Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4×#95 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,11,14,17)(12,16,20,24,22,19,21,23);(2,6,8,7)(9,11)(10,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23);(9,12,14,21,15,22,13,20)(10,16,17,23,11,19,18,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=h,g^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12656);