Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8977 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8,11,12)(3,13,14,20,16,31,15,30)(5,18)(9,24,22,32,26,27,29,25)(10,19,28,23)(17,21);(1,2,7,8,6,11,4,12)(3,9,14,22,16,26,15,29)(5,19,21,10,18,23,17,28)(13,24,20,32,31,27,30,25);(1,3)(2,9)(4,15)(5,20)(6,16)(7,14)(8,22)(10,27)(11,26)(12,29)(13,17)(18,30)(19,32)(21,31)(23,25)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214124, #72214153, #72214308 |
| maximal normal subgroup of | #72214124, #72214153, #72214308 |
| maximal quotient of | #72216469, #72216591, #72216594 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44066);