Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1081 conjugacy classes of subgroups; only the 595 normal subgroups are shown.
This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 595 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) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,7)(8,10,12,11)(13,14,17,20)(15,19,16,18);(5,6,9,7)(8,11,12,10)(13,15,17,16)(14,18,20,19);(1,2,3,4)(6,7)(10,11)(13,16,17,15)(14,19,20,18);(5,7,9,6)(8,10,12,11);(5,8,9,12)(6,10,7,11)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=gh,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 | #36593⋊C3 |
| maximal subgroup of | #98228, #98256, #130564, … |
| maximal normal subgroup of | #98228, #98256, #130564, … |
| maximal quotient of | #519060, #519096, #519397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29581);