Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
251 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10)(11,15,20,29)(12,22)(13,25)(14,17)(16,23)(18,26)(19,31,32,40)(21,35,34,28)(24,36)(27,38,39,33)(30,37);(1,2,6,8,4,7,3,5)(9,11,14,20)(10,15)(12,19,24,32)(13,21,26,34)(16,27)(17,29)(18,28)(22,31)(23,33,37,38)(25,35)(30,39)(36,40);(9,12,13,23,14,24,26,37)(10,16,18,22,17,30,25,36)(11,19,21,33,20,32,34,38)(15,27,28,31,29,39,35,40)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131842, #131846, #136509, … |
| maximal normal subgroup of | #131842, #131846, #136509, … |
| maximal quotient of | #102032, #102036, #102037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2744);