Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512565 |
| 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×D4 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | #60390 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×#41 ◁ #3500 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1444 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1444 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:
(1,2,6,12,8,3,13,17)(4,9,19,26,20,14,28,30)(5,10,22,11,7,15,23,16)(18,24,31,29,21,27,32,25);(1,3)(2,8)(4,14)(5,15)(6,16)(7,10)(9,20)(11,13)(12,22)(17,23)(18,27)(19,29)(21,24)(25,28)(26,32)(30,31);(1,4,8,20)(2,9,3,14)(5,21,7,18)(6,19,13,28)(10,27,15,24)(11,25,16,29)(12,26,17,30)(22,32,23,31);(1,5,8,7)(2,10,3,15)(4,18,20,21)(6,22,13,23)(9,24,14,27)(11,17,16,12)(19,31,28,32)(25,26,29,30)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,f^b=fgi,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, 419769);