Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(1,2,3,4)(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(5,6,7,8)(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=g,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #33308⋊C3 |
| maximal subgroup of | #141095, #144085, #144087, … |
| maximal normal subgroup of | #141095, #144085, #144087, … |
| maximal quotient of | #352907, #352911, #352913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26296);