Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
986 conjugacy classes of subgroups; only the 409 normal subgroups are shown.
This group has 986 conjugacy classes of subgroups — too many to draw. Only its 409 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) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(17,19,20,24)(18,21,22,23);(3,7)(6,8)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24);(1,3,4,7)(2,6,5,8)(18,22)(21,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|b^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #36694⋊C3, #72266665 |
| maximal subgroup of | #97005, #131203, #131209, … |
| maximal normal subgroup of | #97005, #131203, #131209, … |
| maximal quotient of | #544315, #544321, C4×#2819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29682);