Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,9,11,16)(4,12,14,8)(6,13,15,10)(22,28)(29,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(2,7)(6,15)(8,12)(9,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #250780, #250782, #250799, … |
| maximal normal subgroup of | #250780, #250782, #250799, … |
| maximal quotient of | #167044, #167050, #167107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11263);