Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
346 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(2,5,6,7)(9,10)(11,19,22,15)(12,23,25,32)(13,18)(14,17)(16,24,30,26)(20,35)(21,34,39,29)(27,31)(28,36)(33,38)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,19)(12,20,25,37)(13,21,27,39)(16,28,30,33)(18,29,31,34)(23,40,32,35)(24,36,26,38);(3,8)(5,7)(9,12)(10,16)(11,20)(13,26)(14,25)(15,28)(17,30)(18,32)(19,33)(21,38)(22,37)(23,31)(24,27)(29,35)(34,40)(36,39)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #94535, #94537, #152563, … |
| maximal normal subgroup of | #94535, #94537, #152563, … |
| maximal quotient of | #96981, #96984, #96991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1480);