Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
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 | |
171 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14,16,21,30,38,41,34,18,23,31,33,17,22,28,19)(15,24,25,39,32,44,43,36,27,40,42,35,26,37,29,20);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(16,28)(17,30)(20,24)(21,22)(23,34)(25,29)(26,32)(31,41)(33,38)(35,44)(36,40)(37,39)(42,43);(2,4)(13,15,18,27)(14,20,23,36)(16,29,31,43)(17,32,30,26)(19,24,34,40)(21,37,33,44)(22,39,38,35)(25,41,42,28)Polycyclic presentation:
2^8|a^2=d,b^2=deg,c^2=h,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,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 | C2×#13529, #499527, #499529, … |
| maximal normal subgroup of | C2×#13529, #499527, #499529, … |
| maximal quotient of | #146193, #146211, #146217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6517);