Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,15,20)(11,21,12,24)(13,27,32,40)(14,33,30,19)(16,25,17,22)(18,36,39,31)(23,38,26,37)(28,35,29,34);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,20,17)(13,28,32,29)(14,26,30,23)(18,37,39,38)(19,35,33,34)(21,40,24,27)(22,36,25,31);(6,8)(9,12,15,11)(10,17,20,16)(13,29,32,28)(14,23,30,26)(18,38,39,37)(19,34,33,35)(21,27,24,40)(22,31,25,36)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125640, #125641, #125663, … |
| maximal normal subgroup of | #125640, #125641, #125663, … |
| maximal quotient of | #142631, #142709, #142827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6318);