Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150123 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 98 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)(5,6,9,15,31,23,28,35,11,16,8,25,27,24,34,36,12,17,29,26,13,32,33,20,19,22,30,14,10,7,18,21);(5,7,19,23,12,24,11,32)(6,13,16,27,17,31,22,10)(8,26,9,14,30,15,29,25)(18,21,33,35,34,36,28,20);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,d^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319425, #72319620, D4×#1419 |
| maximal normal subgroup of | #72319425, #72319620, D4×#1419 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60186);