Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,16,13,24,35,20,11,17,30,14,33,25,36,21,12,18,31,15,28,34,29,27,32,23,8,26,10,7,19,22);(1,2,3,4)(5,7,11,34,12,25,32,24)(6,13,23,33,18,28,17,10)(8,16,9,26,30,15,31,14)(19,22,35,27,36,21,29,20);(5,6)(7,13)(8,27)(9,21)(10,24)(11,23)(12,18)(14,35)(15,19)(16,36)(17,32)(20,30)(22,31)(25,28)(26,29)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=egi,f^b=fg,g^b=gh,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286674, #72319425, #72319593, … |
| maximal normal subgroup of | #72286674, #72319425, #72319593, … |
| maximal quotient of | #72286690, #72321050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59744);