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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,14,19)(11,20,23,38)(12,16,27,34)(13,28,26,18)(15,30,32,21)(17,35,33,25)(22,39,29,36)(24,37,40,31);(2,4)(5,6)(7,8)(9,11,14,23)(10,15,19,32)(12,24,27,40)(13,29,26,22)(16,20,34,38)(17,36,33,39)(18,37,28,31)(21,35,30,25);(6,8)(10,16)(12,25)(13,26)(15,20)(17,18)(19,34)(21,24)(22,29)(27,35)(28,33)(30,40)(31,39)(32,38)(36,37)Polycyclic presentation:
2^8|a^2=h,b^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 | C2×#13329, #498844, #498846, … |
| maximal normal subgroup of | C2×#13329, #498844, #498846, … |
| maximal quotient of | #142629, #142630, #142708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6317);