Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,17,12,16,21,15)(11,18,20,24,19,14,22,23);(2,5)(6,7)(9,11,13,20,12,19,21,22)(10,14,17,23,16,18,15,24);(10,15)(11,19)(13,21)(14,23)(16,17)(18,24);(1,3,4,8)(2,6,5,7)(10,16)(14,18)(15,17)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25316, #6448073, #6448074, … |
| maximal normal subgroup of | C2×#25316, #6448073, #6448074, … |
| maximal quotient of | #286655, #286661, #286667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18304);