Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #72216494 (order 1024) · Inn(G) = C2×D8 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C8×D8 |
| prime radical | #508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,8,13,25,11,9,14,5,24,7,3,15,17)(4,18,20,32,22,16,30,31,23,10,19,27,21,28,29,26);(1,3)(2,8)(4,16)(5,11)(6,17)(7,14)(9,13)(10,22)(12,25)(15,24)(18,21)(19,26)(20,31)(23,28)(27,30)(29,32);(1,4,6,20,8,22,25,30,9,23,5,19,7,21,15,29)(2,10,12,27,13,28,11,26,14,18,24,32,3,16,17,31)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fh,d^b=df,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93759, #151086, #151103, … |
| maximal normal subgroup of | #93759, #151086, #151103, … |
| maximal quotient of | #122944, #122945, #122946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5035);