Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,13,9,14,19,26,10,11,18,22,20,23,30,33)(7,12,15,24,16,25,28,34,17,21,27,31,29,32,35,36);(1,2,3,4)(6,11)(12,21)(14,23)(25,32);(5,7,9,16,10,17,20,29)(6,12,14,25,11,21,23,32)(8,15,19,28,18,27,30,35)(13,24,26,34,22,31,33,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,e^2=h,h^2=i,b^a=bd,d^a=dg,g^a=gi,e^b=eg,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279512, #72279588, #72279944, … |
| maximal normal subgroup of | #72279512, #72279588, #72279944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34153);