Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | Aut(G) = #72286380 (order 4096) · Inn(G) = #1316 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1316 |
| commutator | C24 | C2×C8 |
| Frattini, cosocle | #407 | C22 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16);(2,6)(3,9)(4,11)(7,12)Polycyclic presentation:
2^8|a^2=d,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=de,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94518, #94519, #94558, … |
| maximal normal subgroup of | #94518, #94519, #94558, … |
| maximal quotient of | #93194, #93198, #93671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 328);