Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1889 |
| prime radical | #537 | C2×C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(2,5,6,7)(9,10,14,20,15,21,12,18)(11,22,26,16,23,32,24,33)(13,19,27,35,30,17,29,36)(25,37,38,34,28,40,39,31);(1,2,3,7,4,6,8,5)(9,11)(10,16)(12,26)(13,28)(14,24)(15,23)(17,34)(18,32)(19,31)(20,22)(21,33)(25,30)(27,38)(29,39)(35,40)(36,37);(10,17,21,19)(11,23)(16,31,33,34)(18,36,20,35)(22,37,32,40)(24,26)(25,28)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,f^c=fh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44283);