Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,13,16,14,17,12,18)(11,15,20,23,19,24,21,22);(1,2,3,5,4,6,7,8)(9,11)(10,15,17,24)(12,20)(13,21)(14,19)(16,22,18,23);(10,16)(12,13)(15,22)(17,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13614);