Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,14,11,16,10,15)(7,12,18,27,19,28,25,32)(8,20,22,31,24,30,23,13)(17,29,21,35,33,36,34,26);(1,2,4,3)(5,7)(6,12)(8,21)(9,18)(10,25)(11,19)(13,29)(14,27)(15,32)(16,28)(17,23)(20,35)(22,33)(24,34)(26,30)(31,36);(5,8,11,24)(6,13,16,31)(7,17,19,33)(9,23,10,22)(12,26,28,35)(14,30,15,20)(18,34,25,21)(27,36,32,29);(1,3,4,2)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #272163, #272165, #272180, … |
| maximal normal subgroup of | #272163, #272165, #272180, … |
| maximal quotient of | #276957, #276964, #276969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17760);