Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,4,7,6,3)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,23)(21,24);(1,3,6,7,4,8,5,2)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,23)(20,24);(2,7)(3,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ci,d^b=dh,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, 12807);