Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| 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 | |
453 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 453 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)(9,10,12,20,15,21,13,18)(11,22,25,16,26,33,23,34)(14,19,28,17,30,35,27,36)(24,37,38,40,29,31,39,32);(1,2,3,7,4,6,8,5)(9,11)(10,16)(12,23)(13,25)(14,29)(15,26)(17,31)(18,33)(19,32)(20,22)(21,34)(24,30)(27,39)(28,38)(35,40)(36,37);(9,12,15,13)(10,17,21,36)(11,23,26,25)(14,28,30,27)(16,31,34,37)(18,35,20,19)(22,32,33,40)(24,39,29,38)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44270);