Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C22×#120×#392 (order 8192) · Inn(G) = C24 · Out(G) = C2×#35680 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C24 |
| commutator, socle | C22 | C22×C42 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,23,24);(1,2,3,6,4,5,7,8);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(10,15)(11,17)(16,22)(18,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,f^2=h,d^a=dg,c^b=ch,d^c=dg100 conjugacy classes — view the full character table.
| maximal subgroup of | #128638, #184033, #184034, … |
| maximal normal subgroup of | #128638, #184033, #184034, … |
| maximal quotient of | #180817, #180841, #180856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10325);