Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138660 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9221 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 362 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:
(1,2,6,12,5,13,22,20,7,10,24,11,9,4,15,21)(3,14,17,30,19,28,23,26,16,31,32,29,18,8,25,27);(1,3,15,32,9,19,24,25,7,16,22,17,5,18,6,23)(2,8,11,29,13,31,21,26,10,28,12,30,4,14,20,27);(1,4)(2,9)(3,14)(5,10)(6,12)(7,13)(8,19)(11,24)(15,20)(16,31)(17,26)(18,28)(21,22)(23,30)(25,29)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214066, #72214153, #72214217 |
| maximal normal subgroup of | #72214066, #72214153, #72214217 |
| maximal quotient of | #72216471, #72227245, #72227248 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45856);