Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 421 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:
(2,4)(5,6,10,16,11,14,9,15)(7,12,18,28,19,27,25,32)(8,20,23,30,24,13,22,31)(17,33,21,36,34,26,35,29);(1,2,3,4)(5,7)(6,12)(8,21)(9,25)(10,18)(11,19)(13,29)(14,27)(15,32)(16,28)(17,22)(20,36)(23,34)(24,35)(26,30)(31,33);(5,8,11,24)(6,13,14,20)(7,17,19,34)(9,23,10,22)(12,26,27,33)(15,31,16,30)(18,35,25,21)(28,36,32,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11059, #357707, #358179, … |
| maximal normal subgroup of | C2×#11059, #357707, #358179, … |
| maximal quotient of | #115710, #115752, #115754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4047);