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 | |
163 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,37)(24,32)(25,33)(26,34)(27,35)(28,36)(29,39)(30,40)(31,41)(38,43)(42,44);(6,9)(10,12)(13,15,16,24)(14,19,20,32)(17,23,27,41)(18,30,28,26)(21,37,35,31)(22,40,36,34)(25,43,42,29)(33,38,44,39);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(15,23)(17,29)(18,28)(19,31)(21,38)(24,41)(25,26)(27,43)(30,42)(32,37)(33,40)(34,44)(35,39)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13483, #499345, #499347, … |
| maximal normal subgroup of | C2×#13483, #499345, #499347, … |
| maximal quotient of | #146007, #146020, #146039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6471);