Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
480 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 79 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)(5,6,9,10,8,7,11,12)(14,17,18,19);(9,11)(10,12)(13,14)(15,19)(16,18)(17,20);(2,4)(5,7)(6,8)(9,10)(11,12)(15,20)(17,19)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fg40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8506⋊C3 |
| maximal subgroup of | #152551, #152562, #153129, … |
| maximal normal subgroup of | #152551, #152562, #153129, … |
| maximal quotient of | #97000, #97005, #97174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1494);