Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | #2656 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
472 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(9,12)(13,14,17,20)(15,19,16,18);(2,4)(5,7)(6,9)(8,11)(10,12)(14,18)(15,16)(19,20)Polycyclic presentation:
2^8|b^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93215, #125666, #125667, … |
| maximal normal subgroup of | #93215, #125666, #125667, … |
| maximal quotient of | #102431, #102443, #102459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2952);