Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9061 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19661 | 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 | |
473 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 473 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:
(1,2,7,13,6,17,5,24,9,12,25,16,8,3,14,11)(4,18,21,31,20,30,26,32,23,10,27,29,22,15,19,28);(1,3,8,2,9,17,6,12)(4,15,20,10,23,30,22,18)(5,24,7,16,14,11,25,13)(19,28,21,29,26,32,27,31);(1,4)(2,10)(3,15)(5,21)(6,22)(7,26)(8,20)(9,23)(11,31)(12,18)(13,28)(14,27)(16,32)(17,30)(19,25)(24,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=ghi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214114, #72214383, #72214466, … |
| maximal normal subgroup of | #72214114, #72214383, #72214466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44760);