Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512579 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C4⋊Q8 | C24 |
| prime radical | #60388 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1486 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1486 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 4 generators on 32 points:
(1,2,6,13,8,15,5,11)(3,16,20,9,21,27,32,28)(4,10,22,12,7,14,23,29)(17,30,18,25,24,26,19,31);(1,3)(2,9)(4,17)(5,18)(6,19)(7,24)(8,21)(10,25)(11,26)(12,27)(13,30)(14,31)(15,28)(16,29)(20,22)(23,32);(1,4,8,7)(2,10,15,14)(3,17,21,24)(5,23,6,22)(9,25,28,31)(11,29,13,12)(16,30,27,26)(18,32,19,20);(1,5,8,6)(2,11,15,13)(3,18,21,19)(4,22,7,23)(9,26,28,30)(10,12,14,29)(16,25,27,31)(17,20,24,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,f^e=fhi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419783);