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 | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6,7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(12,20)(15,22)(18,24)(21,23);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,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, 6250504);