Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 593 conjugacy classes of subgroups — too many to draw. Only its 175 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)(8,22,25,30)(9,15)(10,14)(11,16)(12,19)(13,24,29,23)(18,32,35,26)(20,28)(21,27)(31,34,36,33);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,24,34,25,35,23,33)(13,26,30,36,29,32,22,31);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129789, #129790, #137980, … |
| maximal normal subgroup of | #129789, #129790, #137980, … |
| maximal quotient of | #105634, #105635, #105638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3761);