Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#550 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1261 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1261 conjugacy classes of subgroups — too many to draw. Only its 503 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,10,8)(7,11,12,9)(15,19)(17,20);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14,16,18)(15,17,19,20);(1,3,4,2)(6,8)(7,12)(13,15)(14,17)(16,19)(18,20);(5,8,10,6)(7,11,12,9);(5,9,10,11)(6,7,8,12)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fh,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #36583⋊C3 |
| maximal subgroup of | #98262, #130556, #130563, … |
| maximal normal subgroup of | #98262, #130556, #130563, … |
| maximal quotient of | #518986, #519054, #519096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29571);