Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| 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 | |
355 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,4,3)(5,6,9,13)(7,15)(8,20,23,34)(10,25,24,14)(11,26)(12,30,32,21)(16,33)(17,28)(18,35)(19,31)(22,27)(29,36);(1,3,4,2)(8,21)(10,24)(12,20)(14,25)(15,33)(16,19)(17,36)(22,26)(23,30)(27,29)(31,35)(32,34);(5,7)(6,11)(8,22)(9,18)(10,16)(12,31)(13,28)(14,27)(15,34)(17,30)(19,24)(20,35)(21,26)(23,36)(25,29)(32,33)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224799, #72224804, #72224811, … |
| maximal normal subgroup of | #72224799, #72224804, #72224811, … |
| maximal quotient of | #72242501, #72242509 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43558);