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 | #127 | C24 |
| prime radical | #60391 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×#41 ◁ #3499 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1437 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1437 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:
(1,2,7,12)(3,14,17,22)(4,9,20,28)(5,10,24,29)(6,26,8,13)(11,25,30,23)(15,31,16,21)(18,32,27,19);(1,3)(2,4)(5,15)(6,16)(7,27)(8,9)(10,19)(11,14)(12,31)(13,22)(17,24)(18,25)(20,29)(21,30)(23,28)(26,32);(1,4,8,22)(2,9,13,3)(5,21,25,19)(6,14,7,20)(10,15,30,18)(11,27,29,16)(12,28,26,17)(23,32,24,31);(1,5,8,25)(2,10,13,30)(3,15,9,18)(4,19,22,21)(6,23,7,24)(11,12,29,26)(14,31,20,32)(16,28,27,17)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^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.
| maximal subgroup of | #72338308, #72338314, #72338456 |
| maximal normal subgroup of | #72338308, #72338314, #72338456 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419762);