Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
585 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(1,3,4,7)(2,5,6,8);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|b^2=g,c^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246347, #246348, #246363, … |
| maximal normal subgroup of | #246347, #246348, #246363, … |
| maximal quotient of | #165884, #165895, #165922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11131);