Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137558 |
| 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:
(1,2,7,12,25,13,5,22,8,11,26,16,6,3,14,10)(4,17,20,31,24,29,32,30,21,9,23,28,19,15,18,27);(1,3,6,2,8,13,25,11)(4,15,24,9,21,29,19,17)(5,22,7,16,14,10,26,12)(18,27,20,28,32,30,23,31);(1,4)(2,9)(3,15)(5,23)(6,24)(7,18)(8,21)(10,28)(11,17)(12,30)(13,29)(14,20)(16,27)(19,25)(22,31)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=gh,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=eg,f^b=fg,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, #72214645, #72214714, … |
| maximal normal subgroup of | #72214114, #72214645, #72214714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44756);