Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62816 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3160 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)(9,10,14,18,15,11,13,17)(12,16,22,24,20,19,21,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,20)(21,24)(22,23);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24);(12,20)(16,19)(21,22)(23,24);(9,13,15,14)(10,17,11,18)(12,21,20,22)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=i,g^2=h,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=dh122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250958);