Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132253 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C2×H(3,𝔽4) | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#556 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
688 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5,6,7)(9,10,14,20,15,21,12,18)(11,22,23,16,25,32,29,33)(13,17,26,34,28,19,30,35)(24,37,40,31,27,38,39,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,23)(13,27)(14,29)(15,25)(17,31)(18,22)(19,36)(20,32)(21,33)(24,28)(26,40)(30,39)(34,38)(35,37);(3,8)(5,7)(10,17)(12,14)(13,28)(16,31)(18,35)(19,21)(20,34)(22,37)(23,29)(24,27)(32,38)(33,36)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,e^c=ei,f^c=fg,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39469);