Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1879 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8947 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(2,7,10,23,12,24,11,19)(3,13,15,20,17,14,16,32)(4,18)(8,25)(22,31)(26,27);(1,2,8,7,21,11,4,19,6,12,25,24,5,10,18,23)(3,9,14,22,16,30,20,26,17,29,13,31,15,28,32,27);(1,3)(2,9)(4,20)(5,15)(6,17)(7,22)(8,14)(10,28)(11,30)(12,29)(13,25)(16,21)(18,32)(19,26)(23,27)(24,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225460, #72225465, #72225466 |
| maximal normal subgroup of | #72225460, #72225465, #72225466 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43878);