Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18,13,17,20,24)(11,19,12,16,21,15,22,23);(1,2)(3,4)(9,11)(10,15)(12,14)(13,21)(16,24)(17,19)(18,23)(20,22);(6,8)(9,12)(10,16)(11,20)(13,22)(14,21)(15,18)(17,23)(19,24);(2,4)(9,12)(10,16)(11,20)(13,22)(14,21)(15,18)(17,23)(19,24);(11,21)(12,22)(15,19)(16,23)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^b=ei,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286201);