Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1304 |
| commutator | C22×C4 | C42 |
| Frattini, prime radical, cosocle | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9)(3,11,13,28)(4,15,16,26)(6,10,19,27)(7,20,21,30)(8,18,24,17)(12,22,29,25)(14,23,31,32);(1,3)(2,7)(4,12)(5,13,19,31)(6,14)(8,23)(9,25,27,21)(10,22)(11,15)(16,29,18,20)(17,30)(24,28,26,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gh,c^b=cg,d^b=dfh,f^b=fh,d^c=dh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7313, #128042, #128079, … |
| maximal normal subgroup of | C2×#7313, #128042, #128079, … |
| maximal quotient of | #93109, #93110, #93115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 301);