Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106017 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
708 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7,8,5)(4,6)(9,10)(11,21,25,31)(12,26,28,34)(13,18)(14,19)(15,20)(16,23,32,24)(17,27,33,29)(22,36)(30,38)(35,39)(37,40);(9,11)(10,16,20,32)(12,22)(13,23)(14,24)(15,25)(17,30,33,36)(18,21,19,31)(26,35,34,40)(27,37)(28,38)(29,39);(1,3)(2,5)(4,8)(6,7)(9,12,13,27,15,28,14,29)(10,17,18,26,20,33,19,34)(11,22,23,37,25,38,24,39)(16,30,21,35,32,36,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,f^b=fi,e^c=ei,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15676);