Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #418 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #7895 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(1,2,5,12,7,13,20,11,8,14,4,9,6,22,21,24)(3,10,15,25,17,27,31,29,18,28,19,23,16,32,30,26);(2,9,13,24,14,12,22,11)(10,25,27,29,28,23,32,26);(1,3)(2,10)(4,19)(5,15)(6,16)(7,17)(8,18)(9,23)(11,29)(12,25)(13,27)(14,28)(20,31)(21,30)(22,32)(24,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=cgh,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320359, #72320704, #72320735 |
| maximal normal subgroup of | #72320359, #72320704, #72320735 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34046);