Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,3,6,4,5,7,8)(9,10,14,20,15,18,30,36,16,21,28,38,12,27,13,19)(11,22,25,35,23,37,24,32,26,39,29,17,31,33,40,34);(2,5)(3,7)(9,11)(10,17)(12,23)(13,29)(14,24)(15,31)(16,26)(18,32)(19,37)(20,22)(21,35)(25,30)(27,34)(28,40)(33,36)(38,39);(2,5)(6,8)(9,12,16,15)(10,18,21,27)(11,23,26,31)(13,28,30,14)(17,32,35,34)(19,20,36,38)(22,33,39,37)(24,29,40,25)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,e^2=fg,g^2=h,b^a=bd,c^a=cfh,d^a=df,d^b=dg,e^b=efg,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131050, #131051, #131053, … |
| maximal normal subgroup of | #131050, #131051, #131053, … |
| maximal quotient of | #124253, #124454, #124455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4948);