Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | Aut(G) = #72309372 (order 4096) · Inn(G) = #447 · Out(G) = C22×M16 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,8)(10,16)(11,20)(12,23)(13,14)(17,31)(18,35)(19,30)(21,37)(22,27)(24,28)(25,26)(29,32)(33,40)(34,38);(1,2)(3,5)(4,6)(7,8)(9,10,15,19)(11,17,22,34)(12,24,26,35)(13,18,14,28)(16,25,30,23)(20,32,27,33)(21,38,37,31)(29,39,40,36);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,21)(13,27)(14,20)(15,22)(16,29)(18,33)(19,34)(23,36)(24,38)(25,39)(26,37)(28,32)(30,40)(31,35)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12165, #484636, #484638, … |
| maximal normal subgroup of | C2×#12165, #484636, #484638, … |
| maximal quotient of | #123172, #123175, #123178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5153);