Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,15)(12,13)(16,18)(17,21)(19,22)(23,24);(9,10,12,15,14,17,13,21)(11,18,19,16,20,23,22,24);(9,11)(10,16)(12,19)(13,22)(14,20)(15,23)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23633);