Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,22)(18,23)(20,24);(10,16)(14,21)(15,22)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251506);