Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,27,30,23)(20,24,22,26)(28,32,29,31);(2,5)(6,8)(17,19,21,30)(18,23,25,27)(20,29,22,28)(24,32,26,31);(9,11,12,16)(10,13,14,15)(19,28)(20,22)(23,31)(24,26)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,24,25,26)(19,29,30,28)(23,32,27,31)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23785, #6432538, #6432544, … |
| maximal normal subgroup of | C2×#23785, #6432538, #6432544, … |
| maximal quotient of | #358526, #358629, #358630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16773);