Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | Aut(G) = C2×#72225051 (order 2048) · Inn(G) = #357 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #357 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
51 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,13,18,28,25,37,38,16,20,33,32,17,23,30,36,14,19,12,26,29,35,40,39,34,24,31,27,15,11,21,22);(1,2,4,6)(3,8,7,5)(9,11,14,23)(10,17,19,15)(12,27,13,32)(16,35,34,25)(18,31,26,33)(20,28,24,29)(21,36,30,22)(37,38,40,39)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=efgh,f^2=gh,g^2=h,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fg,g^b=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7452, #150958, #150959, … |
| maximal normal subgroup of | C2×#7452, #150958, #150959, … |
| maximal quotient of | #93931, #93932, #94352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 440);