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 | #60390 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×#41 ◁ #3500 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1456 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:
(1,2,4,9)(3,7,13,23)(5,18,19,15)(6,11,17,25)(8,20,10,16)(12,21,26,32)(14,30,31,28)(22,27,24,29);(1,2)(3,7)(4,15)(5,18)(6,11)(8,17)(9,16)(10,20)(12,21)(13,28)(14,30)(19,25)(22,26)(23,29)(24,27)(31,32);(1,3)(2,7)(4,13)(5,14)(6,12)(8,22)(9,23)(10,24)(11,21)(15,28)(16,29)(17,26)(18,30)(19,31)(20,27)(25,32);(3,12)(7,21)(13,26)(14,27)(22,28)(23,32)(24,30)(29,31)Polycyclic presentation:
2^9|a^2=f,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.
| maximal subgroup of | #72306285, #72306359, #72306360, … |
| maximal normal subgroup of | #72306285, #72306359, #72306360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419775);