Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C22×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
355 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 161 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3,4,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,21)(22,24);(2,6)(3,7)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(1,3,5,6,4,7,8,2)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(10,16)(11,18)(14,17)(19,24)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,c^b=ch,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #135037, #135040, #135070, … |
| maximal normal subgroup of | #135037, #135040, #135070, … |
| maximal quotient of | #182421, #183624, #183931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10883);