Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, prime radical, cosocle | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 2 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19,21,30)(12,23,25,16)(13,17)(14,18)(15,20)(22,34,37,40)(24,29)(26,33,35,31)(27,32)(28,36)(38,39);(1,3)(2,5,6,7)(4,8)(9,11,13,22,14,21,27,37)(10,15,17,28,18,29,32,39)(12,24,26,38,25,20,35,36)(16,30,31,40,23,19,33,34)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,e^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,c^b=cg,d^b=dfh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7309, #128036, #128037, … |
| maximal normal subgroup of | C2×#7309, #128036, #128037, … |
| maximal quotient of | #93111, #93112, #93115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 297);