Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 147456 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3483 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17,13,12,16,21)(11,18,20,24,19,15,22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24);(3,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24);(2,6)(5,7)(11,19)(15,18)(20,22)(23,24);(9,12,14,21,13,10,16,17)(11,18,20,24,19,15,22,23)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=dfh,c^b=cfh,d^b=dh,d^c=dh100 conjugacy classes — view the full character table.
| Fitting subgroup of | #11710305, #11710306 |
| maximal subgroup of | #129594, #129654, #210190, … |
| maximal normal subgroup of | #129594, #129654, #210190, … |
| maximal quotient of | #6370400, #6370507, #6377482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53296);