Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,2,4,6)(3,7,8,5)(9,10,15,19)(11,16,14,29)(12,22,24,38)(13,25,28,39)(17,32,34,23)(18,21,36,27)(20,37,30,35)(26,31,40,33);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,34,33)(18,25,36,39)(22,37,38,35)(23,26,32,40);(9,12)(10,17)(11,20)(13,26)(14,30)(15,24)(16,31)(18,35)(19,34)(21,32)(22,25)(23,27)(28,40)(29,33)(36,37)(38,39);(3,8)(5,7)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,34,33)(18,25,36,39)(22,37,38,35)(23,26,32,40)Polycyclic presentation:
2^8|a^2=eh,b^2=h,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=de,f^c=fg,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #491716, #491725, #491732, … |
| maximal normal subgroup of | #491716, #491725, #491732, … |
| maximal quotient of | #497381, #497542, #497719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26858);