Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1790 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 659 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21)(12,25,13,18)(14,19)(15,20)(16,22,32,27)(17,26,33,28)(23,30)(24,34)(29,36)(31,37,35,38)(39,40);(9,11,15,24)(10,16,20,32)(12,22,13,27)(14,23,29,39)(17,30,33,40)(18,34,25,21)(19,35,36,31)(26,37,28,38);(9,12)(10,17)(11,22)(13,15)(14,28)(16,30)(18,19)(20,33)(21,31)(23,38)(24,27)(25,36)(26,29)(32,40)(34,35)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36778);