Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,3)(9,10)(11,13)(12,15)(14,16)(17,18,21,24)(19,22,20,23);(5,6,7,8)(18,22)(19,20)(23,24);(1,3,4,2)(10,13)(11,16)(14,15)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=dh,e^b=eh,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279300, #72279409, #72279458, … |
| maximal normal subgroup of | #72279300, #72279409, #72279458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17209);