Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
270 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12)(7,17)(8,20)(9,10)(13,29)(14,21)(15,27)(16,28)(18,22)(19,25)(23,24)(26,31)(30,33)(32,35);(1,2)(3,4)(5,6,11,16)(7,13,19,32)(8,21,24,27)(9,15,10,14)(12,23,28,20)(17,33,25,31)(18,35,22,29)(26,34,30,36);(2,4)(5,7)(6,13)(8,22)(9,25)(10,17)(11,19)(12,26)(14,33)(15,31)(16,32)(18,24)(20,36)(21,29)(23,34)(27,35)(28,30)Polycyclic presentation:
2^8|b^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13320, #498830, #498831, … |
| maximal normal subgroup of | C2×#13320, #498830, #498831, … |
| maximal quotient of | #142238, #142249, #142285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6308);