Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,22)(3,12,16,13,17,26,15,8)(4,18,20,24,21,31,19,9)(14,27,29,32,30,23,28,25);(1,3)(2,8,11,13)(4,14)(5,15)(6,16)(7,17)(9,23,24,27)(10,12,22,26)(18,25,31,32)(19,28)(20,29)(21,30);(1,2)(3,13)(4,9)(5,10)(6,22)(7,11)(8,17)(12,16)(14,27)(15,26)(18,19)(20,31)(21,24)(23,30)(25,29)(28,32);(1,4,6,20,7,21,5,19)(2,9,22,31,11,24,10,18)(3,14,16,29,17,30,15,28)(8,23,26,32,13,27,12,25)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #502201, #502202, #502215, … |
| maximal normal subgroup of | #502201, #502202, #502215, … |
| maximal quotient of | #288572, #288576, #288578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18435);