Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
811 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 811 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,21)(12,24)(13,18)(14,19)(15,20)(16,22)(17,23);(1,2,3,4)(6,8)(9,11,13,22,15,23,14,12)(10,16,19,21,20,24,18,17);(9,12,14,23,15,22,13,11)(10,17,18,24,20,21,19,16);(9,13,15,14)(10,18,20,19)(11,22,23,12)(16,17,24,21)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #124069, #124070, #124250, … |
| maximal normal subgroup of | #124069, #124070, #124250, … |
| maximal quotient of | #204257, #204274, #204291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14212);