Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137580 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9064 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| 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 | |
409 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,7,12,9,17,5,11,6,13,25,31,8,3,14,16)(4,18,21,27,23,10,26,28,20,15,24,32,22,29,19,30);(1,3,8,2,6,17,9,13)(4,15,23,10,20,18,22,29)(5,16,7,12,14,11,25,31)(19,27,21,30,26,32,24,28);(1,4)(2,10)(3,15)(5,24)(6,20)(7,19)(8,23)(9,22)(11,30)(12,27)(13,29)(14,21)(16,28)(17,18)(25,26)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=g,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214114, #72214352, #72214424, … |
| maximal normal subgroup of | #72214114, #72214352, #72214424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44777);