Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,14,9,23)(7,12,18,27,19,26,17,33)(8,20,21,13,22,29,24,28)(16,30,31,25,32,36,34,35);(2,3)(5,7,10,18,11,19,9,17)(6,12,15,27,14,26,23,33)(8,16,21,31,22,32,24,34)(13,25,29,36,28,35,20,30);(5,8)(6,13)(7,16)(9,24)(10,21)(11,22)(12,25)(14,28)(15,29)(17,34)(18,31)(19,32)(20,23)(26,35)(27,36)(30,33);(1,3,4,2)(6,14)(12,26)(13,28)(15,23)(20,29)(25,35)(27,33)(30,36)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23017, #6381900, #6381902, … |
| maximal normal subgroup of | C2×#23017, #6381900, #6381902, … |
| maximal quotient of | #229622, #229623, #229624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16005);