Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | Aut(G) = #72252039 (order 2048) · Inn(G) = H(3,𝔽4) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2684 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 97 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,15)(9,20)(14,29)(16,25)(17,24)(19,32)(21,27)(22,28)(23,31)(26,30);(1,3,6,16,7,17,5,15)(2,8,11,24,12,25,10,13)(4,19,20,14,22,30,21,31)(9,26,18,29,27,32,28,23);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,18)(11,28)(12,27)(13,26)(15,30)(16,19)(17,31)(24,32)(25,29);(1,2)(3,13)(4,9)(5,11)(6,10)(7,12)(8,15)(14,26)(16,25)(17,24)(18,21)(19,29)(20,28)(22,27)(23,30)(31,32)Polycyclic presentation:
2^8|b^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=deh,f^c=fh,g^c=gh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153183, #153184, #153185, … |
| maximal normal subgroup of | #153183, #153184, #153185, … |
| maximal quotient of | #345848, #345849, #345850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24269);