Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(10,13)(11,15)(14,16)(17,18)(19,21)(20,23)(22,24);(5,6,7,8)(9,10)(11,16)(12,14)(13,15);(11,15)(13,16)(18,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,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 | #72226850, #72279313, #72279385, … |
| maximal normal subgroup of | #72226850, #72279313, #72279385, … |
| maximal quotient of | #72226846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15530);