Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(6,7)(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(1,2,3,4);(5,7,8,6);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289927);