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×C4 |
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 | C2×#2246 | C2 |
| 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 | |
396 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 396 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,17,11,15,9,21,13,19,23,16,12,18,29,7)(8,22,25,32,26,20,24,14,28,31,35,30,27,34,36,33);(1,2,3,4)(5,7,10,18,11,16,9,19,13,21,23,15,12,17,29,6)(8,20,25,32,26,22,24,33,28,34,35,30,27,31,36,14);(5,8,23,35,11,26,29,36,13,28,10,25,12,27,9,24)(6,14,21,34,18,32,17,31,19,33,7,20,15,30,16,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=eg,d^2=gh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216405, #72216412, #72216416, … |
| maximal normal subgroup of | #72216405, #72216412, #72216416, … |
| maximal quotient of | #72226853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48459);