Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,17)(13,19)(14,21)(15,20)(16,23)(22,30)(24,36)(25,32)(26,37)(27,33)(28,35)(29,38)(31,39)(34,40);(2,5)(6,8)(9,11,15,25,13,24,29,28,16,26,31,40,30,27,14,12)(10,17,21,33,22,34,39,37,23,35,38,36,19,32,20,18);(1,3,4,7)(2,6,5,8)(9,12,14,27,30,40,31,26,16,28,29,24,13,25,15,11)(10,18,20,32,19,36,38,35,23,37,39,34,22,33,21,17);(9,13,16,30)(10,19,23,22)(11,24,26,27)(12,25,28,40)(14,15,29,31)(17,32,35,34)(18,36,37,33)(20,38,39,21)Polycyclic presentation:
2^8|b^2=f,c^2=e,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #137509, #137569, #141277, … |
| maximal normal subgroup of | #137509, #137569, #141277, … |
| maximal quotient of | #494083, #494904, #495116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26685);