Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #390 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
223 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,4,7,5,3,6,8)(9,10)(11,15,20,29)(12,22)(13,26)(14,17)(16,24)(18,23)(19,31,33,40)(21,35,32,28)(25,36)(27,38,39,34)(30,37);(1,3,4,8,5,2,6,7)(9,11,14,20)(10,15)(12,19,25,33)(13,21,23,32)(16,27)(17,29)(18,28)(22,31)(24,34,37,38)(26,35)(30,39)(36,40);(9,12,23,37,14,25,13,24)(10,16,26,36,17,30,18,22)(11,19,32,38,20,33,21,34)(15,27,35,40,29,39,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,f^b=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #136691, #136692, #136694, … |
| maximal normal subgroup of | #136691, #136692, #136694, … |
| maximal quotient of | #101993, #101995, #101998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2657);