Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
520 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,8,12,10)(9,11)(13,14,16,19,17,15,20,18);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20);(5,8,9,10)(6,7,11,12)(16,20)(18,19)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8408, #185927, #185929, … |
| maximal normal subgroup of | C2×#8408, #185927, #185929, … |
| maximal quotient of | #96955, #96956, #97098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1396);