Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,11)(7,9,12,10)(14,17)(15,18)(19,20);(1,2)(3,4)(13,14,16,17)(15,18,19,20);(2,4)(6,9)(7,12)(10,11)(13,15)(14,18)(16,19)(17,20);(15,19)(18,20)Polycyclic presentation:
2^8|a^2=eh,b^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=de,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #166345, #166347, #166361, … |
| maximal normal subgroup of | #166345, #166347, #166361, … |
| maximal quotient of | #327772, #327780, #327788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22627);