Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1667 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #472 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
218 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 218 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:
(1,2,4,3)(6,13)(7,17)(8,9)(10,11)(14,28)(15,29)(16,27)(18,24)(19,20)(21,32)(22,33)(23,25)(30,31)(34,36);(1,3,4,2)(5,6)(7,14)(8,23)(9,13)(10,25)(11,15)(12,16)(17,26)(18,29)(19,34)(20,28)(21,30)(22,31)(24,27)(32,36)(33,35);(5,7,18,32,11,21,9,17,12,22,8,19,10,20,24,33)(6,14,29,36,15,30,13,26,16,31,23,34,25,28,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ehi,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,g^b=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216405, #72216413, #72216414, … |
| maximal normal subgroup of | #72216405, #72216413, #72216414, … |
| maximal quotient of | #72225600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47633);