Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3397 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)(3,4)(9,10,14,19,15,16,13,18)(11,20,22,23,21,24,17,12);(5,6)(7,8)(9,11)(10,12)(13,22)(14,17)(15,21)(16,23)(18,20)(19,24);(2,4)(6,8)(10,16)(11,21)(17,22)(18,19);(2,4)(9,12)(10,17)(11,18)(13,20)(14,24)(15,23)(16,22)(19,21);(9,13,15,14)(10,18,16,19)(11,22,21,17)(12,20,23,24)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,f^b=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407994);