Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(10,15,17,20)(16,23,24,22);(1,2,4,7,6,8,5,3)(9,10,13,15,14,17,12,20)(11,16,18,22,19,24,21,23);(1,3,4,2,6,7,5,8)(9,11)(10,16)(12,21)(13,18)(14,19)(15,22)(17,24)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17832);