Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8375 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61053 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
794 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,6,12,27,40,7,9,26,33,20,34,48,61)(3,13,15,44,16,35,24,55,17,43,50,59,28,51,56,64)(4,18,21,39,22,42,54,63,23,10,25,36,45,37,57,62)(8,29,31,14,32,46,60,19,30,52,58,47,41,49,38,53);(1,3,7,17)(2,8,9,30)(4,19,23,53)(5,24,26,56)(6,28,20,16)(10,35,18,51)(11,38,33,60)(12,41,34,32)(13,42,43,37)(14,45,47,22)(15,48,50,27)(21,49,25,46)(29,54,52,57)(31,40,58,61)(36,44,39,59)(55,62,64,63);(2,9)(8,30)(10,18)(12,34)(13,43)(32,41)(35,51)(37,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,h^b=hi,f^c=fg,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72307079, #72307095, #72307112, … |
| maximal normal subgroup of | #72307079, #72307095, #72307112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37974);