Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
617 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 617 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,6,3,4,8)(9,10)(11,19,23,33)(12,24)(13,28,29,17)(14,18)(15,21)(16,26,32,25)(20,30,36,40)(22,31)(27,39)(34,37)(35,38);(1,3)(2,6)(4,8)(5,7)(9,11,14,23)(10,15,18,31)(12,20,27,36)(13,22,29,21)(16,30,32,40)(17,33,28,19)(24,38,39,34)(25,37,26,35);(1,4,6,5)(2,7,3,8)(9,12,14,27)(10,16,18,32)(11,20,23,36)(13,26,29,25)(15,30,31,40)(17,24,28,39)(19,34,33,38)(21,37,22,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fg,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38737);