Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(11,15)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,3,5,8)(2,6,7,4)(9,10,12,13)(11,14,15,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(9,11,12,15)(10,14,13,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,4,5,6)(2,3,7,8)(10,13)(14,16)Polycyclic presentation:
2^8|a^2=efh,b^2=eh,c^2=fh,d^2=efh,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #123118, #123128, #125808, … |
| maximal normal subgroup of | #123118, #123128, #125808, … |
| maximal quotient of | #385451, #386830, #387121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25683);