Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | |
236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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)(5,6,11,17,21,36,9,28,13,19,10,30,12,18,29,16)(7,20,22,34,31,15,27,33,24,35,26,14,23,32,8,25);(1,2,3,4)(5,7,21,23,13,24,12,31)(6,14,18,25,19,34,36,33)(8,9,26,10,27,29,22,11)(15,17,35,16,32,30,20,28);(5,8)(6,15)(7,9)(10,23)(11,31)(12,22)(13,27)(14,17)(16,25)(18,35)(19,32)(20,36)(21,26)(24,29)(28,33)(30,34)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13039, #494565, #494566, … |
| maximal normal subgroup of | C2×#13039, #494565, #494566, … |
| maximal quotient of | #136705, #136706, #136739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6027);