Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
363 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,11,18,16)(12,19,20,23,21,15,22,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,23)(22,24);(9,12,13,20,14,21,18,22)(10,15,17,24,11,19,16,23);(10,11)(15,19)(16,17)(23,24)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dh,e^a=eg,d^b=dh,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #136310, #136312, #136676, … |
| maximal normal subgroup of | #136310, #136312, #136676, … |
| maximal quotient of | #182474, #183484, #183907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10963);