Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10,11,12)(13,14,16,23,20,24,28,26,18,22,17,21,19,15,25,27);(5,6,7,10,9,11,8,12)(13,15,18,24)(14,20,22,19)(16,21,17,23)(25,27,28,26);(2,4)(5,7,9,8)(6,10,11,12)(13,16,20,28,18,17,19,25)(14,21,24,27,22,23,15,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=gi,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43828);