Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | C22×D16 | C22 |
| 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 | |
306 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 100 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)(5,6,8,14,11,16,10,27)(7,17,20,34,23,12,22,28)(9,24,19,31,26,32,25,15)(13,30,29,18,33,36,35,21);(1,2,3,4);(6,12)(7,18)(8,10)(9,25)(13,31)(14,34)(15,33)(16,17)(19,26)(20,30)(21,23)(22,36)(24,29)(27,28)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=d,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^c=egi,f^c=fi,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225461, #72225508, #72225513, … |
| maximal normal subgroup of | #72225461, #72225508, #72225513, … |
| maximal quotient of | #72225454, #72225461 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59472);