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) = C24 |
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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 119 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)(8,21)(9,15)(10,14)(11,16)(12,19)(13,23)(18,28)(20,31)(22,30)(24,29)(25,27)(26,34)(32,33)(35,36);(5,7,9,19,11,20,10,25)(6,12,14,17,16,27,15,31)(8,18,22,33,24,34,23,35)(13,28,29,36,30,26,21,32);(5,8,9,22,11,24,10,23)(6,13,14,29,16,30,15,21)(7,18,19,33,20,34,25,35)(12,26,17,32,27,28,31,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131395, #131396, #131397, … |
| maximal normal subgroup of | #131395, #131396, #131397, … |
| maximal quotient of | #120169, #120186, #120189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4224);