Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12099 |
| 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 | C22×#41 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 449 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,4,7)(3,5,8,6)(9,10,14,19,15,20,13,28)(11,21,24,35,25,38,29,32)(12,18,26,17,27,36,30,37)(16,31,33,22,34,39,40,23);(2,5)(3,8)(6,7)(9,11,14,24,15,25,13,29)(10,16,19,33,20,34,28,40)(12,23,26,31,27,22,30,39)(17,35,36,38,37,32,18,21);(9,12)(10,17)(11,22)(13,30)(14,26)(15,27)(16,32)(18,28)(19,36)(20,37)(21,33)(23,25)(24,39)(29,31)(34,35)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43203);