Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(2,4)(5,6)(7,17,20,33)(8,21,24,34)(9,25)(10,27)(11,16)(12,19,30,26)(13,23,31,35)(14,18)(15,28)(22,29)(32,36);(1,2,3,4)(5,7,18,35,28,24,9,26,11,20,36,23,10,8,22,19)(6,12,25,21,27,13,14,33,16,30,29,34,15,31,32,17);(5,8,22,35,10,20,36,19,11,24,9,23,28,7,18,26)(6,13,32,34,15,12,29,33,16,31,14,21,27,30,25,17)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13250, #495240, #495241, … |
| maximal normal subgroup of | C2×#13250, #495240, #495241, … |
| maximal quotient of | #140749, #140750, #140784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6238);