Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| 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 | C42 | C24 |
| prime radical | #3171 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
417 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(1,3)(2,7)(4,12)(5,11)(6,13)(8,16)(9,15)(10,14)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,2)(3,7)(4,8)(5,14)(6,9)(10,11)(12,16)(13,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)Polycyclic presentation:
2^8|b^2=gh,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250331, #250333, #250347, … |
| maximal normal subgroup of | #250331, #250333, #250347, … |
| maximal quotient of | #166310, #166311, #166374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12019);