Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | Aut(G) = #72225074 (order 2048) · Inn(G) = #367 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #367 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | #1372 |
| prime radical | C22 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,28,38,8,14,23,42,30,13,6,29)(3,15,18,44,20,46,19,56,21,48,27,45,57,47,58,64)(4,22,16,39,25,59,49,41,26,60,50,40,55,32,24,10)(9,31,33,62,35,63,61,17,37,51,34,53,36,52,43,54);(1,3,16,31,6,19,4,17,30,57,24,52,23,18,55,62,8,21,50,51,28,58,26,54,7,20,49,63,5,27,25,53)(2,9,32,56,11,34,10,15,12,35,22,45,38,43,39,46,14,37,59,64,42,33,41,48,13,36,60,44,29,61,40,47)Polycyclic presentation:
2^8|a^2=d,b^2=ceg,c^2=egh,d^2=f,e^2=gh,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94614, #94615, #94754, … |
| maximal normal subgroup of | #94614, #94615, #94754, … |
| maximal quotient of | #93884, #94028, C2×#7464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 452);