Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×H(3,𝔽4) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1244 conjugacy classes of subgroups; only the 701 normal subgroups are shown.
This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 701 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 — 5 generators on 20 points:
(1,2,4,3)(6,8)(10,11)(14,17)(15,16);(5,6,9,8)(7,11,12,10)(13,14,18,17)(15,20,16,19);(1,2,4,3)(13,15,18,16)(14,19,17,20);(5,7,9,12)(6,10,8,11)(13,16,18,15)(14,20,17,19);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,20,16,19)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=gh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| Fitting subgroup of | #34524⋊C3, #72267665, #72310335, … |
| maximal subgroup of | #97044, #129578, #129585, … |
| maximal normal subgroup of | #97044, #129578, #129585, … |
| maximal quotient of | #513877, #513961, #513976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27512);