Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3341 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(17,18,23,27)(19,28,31,47)(20,25,22,37)(21,33,36,45)(24,39,42,30)(26,32,44,35)(29,46,38,43)(34,40,48,41);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,23,31)(18,24,27,42)(20,29,22,38)(21,34,36,48)(25,40,37,41)(26,43,44,46)(28,45,47,33)(30,32,39,35);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,40,42,41)(26,33,44,45)(28,46,47,43)(30,34,39,48)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12399, #489079, #489083, … |
| maximal normal subgroup of | C2×#12399, #489079, #489083, … |
| maximal quotient of | #125901, #125918, #126056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5387);