Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #394 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
639 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 639 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22,25,13)(9,16)(10,14)(11,15)(12,19)(18,30,33,35)(20,28)(21,29)(23,27,34,36)(24,31,26,32);(2,4)(5,7)(6,12)(8,23)(9,21)(10,19)(11,20)(13,30)(14,28)(15,29)(16,17)(18,24)(22,35)(25,34)(26,33)(27,31)(32,36);(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,31)(15,22)(16,32)(17,30)(19,23)(20,33)(21,34)(28,35)(29,36)Polycyclic presentation:
2^8|a^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93907, #137399, #137400, … |
| maximal normal subgroup of | #93907, #137399, #137400, … |
| maximal quotient of | #111224, #111282, #111283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4026);