Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| 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 | #121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
426 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 426 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) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(6,12)(13,16)(17,18,22,26)(19,27,30,40)(20,24,31,38)(21,33,35,25)(23,34,37,29)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,22,30)(18,23,26,37)(20,28,31,42)(21,34,35,29)(24,36,38,47)(25,40,33,27)(32,46,44,43)(39,48,45,41);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,22,31)(18,24,26,38)(19,28,30,42)(21,32,35,44)(23,36,37,47)(25,39,33,45)(27,41,40,48)(29,43,34,46);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250491, #250494, #250495, … |
| maximal normal subgroup of | #250491, #250494, #250495, … |
| maximal quotient of | #166330, #166340, #166385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11235);