Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132317 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2635 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #2357 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 518 conjugacy classes of subgroups — too many to draw. Only its 76 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)(11,20)(12,25,27,32)(13,18,29,17)(14,19)(15,22,31,23)(16,26)(21,36,39,30)(24,34)(28,33)(35,38)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,22)(14,24)(16,30)(17,20)(18,34)(19,31)(23,29)(25,35)(26,37)(27,39)(28,38)(32,40)(33,36);(3,8)(5,7)(9,12)(10,16,19,33)(11,21)(13,28)(14,27)(15,30,31,36)(17,25,18,32)(20,35,34,40)(22,38)(23,37)(24,39)(26,29)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,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=fgi,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, 39529);