Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10378 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #393 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
180 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,21)(3,11)(4,16)(8,25,17,20)(10,24,22,12)(14,30)(15,28)(18,19)(23,26,29,32)(27,31);(2,8,9,17)(3,11)(7,20,21,25)(10,26,22,32)(12,29,24,23)(14,27)(15,28)(30,31);(1,2,6,10)(3,12,11,21)(4,17,19,32)(5,9,13,22)(7,15,24,28)(8,16,26,18)(14,20,27,23)(25,31,29,30)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=h,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10379, #238314, #238315, … |
| maximal normal subgroup of | C2×#10379, #238314, #238315, … |
| maximal quotient of | #104730, #104731, #104738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3367);