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×#117 | C23 |
| socle | C22 | #1867 |
| prime radical | #392 | Q8 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 386 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:
(1,2)(3,6)(4,7)(5,8)(9,10,14,19,15,20,13,17)(11,21,25,38,22,36,24,39)(12,26,29,35,28,18,30,37)(16,31,32,27,34,40,33,23);(2,6)(3,4)(7,8)(9,11,14,25,15,22,13,24)(10,16,19,32,20,34,17,33)(12,27,29,40,28,23,30,31)(18,36,37,39,26,21,35,38);(2,7)(3,4)(6,8)(10,17,20,19)(11,22)(12,28)(16,32,34,33)(18,35,26,37)(21,38,36,39)(24,25)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=egi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43377);