Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10,13,15,14,17,12,18)(11,16,21,22,19,24,20,23);(5,6,7,8)(10,15)(12,13)(16,22)(17,18)(20,21)(23,24);(1,3,4,2)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,24)(18,23);(10,17)(11,19)(15,18)(20,21)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,b^a=be,d^a=dh,e^b=eh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #135257, #135258, #146890, … |
| maximal normal subgroup of | #135257, #135258, #146890, … |
| maximal quotient of | #183575, #183577, #183579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13442);