Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| 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 | |
198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 198 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,14,19,12,21,28,18,15,20,31,36,29,17,13,30)(11,22,25,34,27,35,40,38,26,37,39,16,23,33,24,32);(2,5)(3,7)(9,11)(10,16)(12,23)(13,25)(14,24)(15,26)(17,32)(18,35)(19,37)(20,34)(21,38)(22,36)(27,29)(28,39)(30,33)(31,40);(2,5)(6,8)(9,12,15,29)(10,17,20,21)(11,23,26,27)(13,14,28,31)(16,32,34,38)(18,19,30,36)(22,35,37,33)(24,39,40,25)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,e^2=fgh,g^2=h,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131050, #131051, #131053, … |
| maximal normal subgroup of | #131050, #131051, #131053, … |
| maximal quotient of | #124250, #124453, #124664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4951);