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 | #124 | C24 |
| prime radical | #60389 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×#41 ◁ #3500 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1465 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:
(1,2,5,10)(3,7,14,24)(4,8,17,19)(6,12,21,27)(9,18,11,20)(13,22,28,23)(15,26,31,29)(16,25,30,32);(1,2)(3,7)(4,8)(5,19)(6,12)(9,21)(10,20)(11,18)(13,22)(14,29)(15,26)(16,25)(17,27)(23,30)(24,28)(31,32);(1,3,6,16)(2,7,12,25)(4,15,18,13)(5,14,21,30)(8,26,11,22)(9,23,19,29)(10,24,27,32)(17,31,20,28);(1,4,6,18)(2,8,12,11)(3,13,16,15)(5,17,21,20)(7,22,25,26)(9,10,19,27)(14,28,30,31)(23,32,29,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^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, 419776);