Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #42201 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 655 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:
(1,2,4,7,5,3,8,6)(9,10)(11,21)(12,25,28,34)(13,19,14,18)(15,20)(16,23,32,29)(17,27)(22,30,37,40)(24,31)(26,33)(35,36)(38,39);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,21)(19,31)(20,32)(25,38)(26,36)(27,39)(28,37)(33,40)(34,35);(4,8)(6,7)(9,12,15,28)(10,17)(11,22,24,37)(13,26,14,27)(16,30)(18,34)(19,25)(20,33)(21,35)(23,36,29,39)(31,38)(32,40)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fgi,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, 38765);