Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 104 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)(6,12,16,15,18,25,17,24)(7,19,21,20,22,35,27,29)(8,23)(9,13)(14,30,32,33)(28,31,36,34);(1,2,3,4)(5,6,13,12,26,17,8,24,11,18,9,25,10,16,23,15)(7,14,29,28,21,30,19,34,22,32,20,36,27,33,35,31);(5,7)(6,14)(8,19)(9,20)(10,27)(11,22)(12,28)(13,29)(15,31)(16,33)(17,30)(18,32)(21,26)(23,35)(24,34)(25,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287284, #72319230, #72319298, … |
| maximal normal subgroup of | #72287284, #72319230, #72319298, … |
| maximal quotient of | #72287112 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43880);