Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#33441 (order 512) · Inn(G) = D4 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | D4 |
| commutator | C4 | C2×C32 |
| Frattini, cosocle | C2×C32 | C22 |
| socle | C2 | C4⋊C32 |
| prime radical | C22 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,6,13,24,37,7,14,25,38,28,40,51,59,8,11,26,34,29,35,52,56,20,36,46,57,47,58,62,64,9,15,27,39,30,41,53,60,17,10,31,32,44,33,54,55,4,21,22,48,23,49,50,63,3,16,18,42,19,43,45,61);(1,3,4,17,9,20,8,7)(2,10,11,16,15,14,21,36)(5,18,22,31,27,46,26,25)(6,19,23,44,30,47,29,28)(12,32,34,42,39,38,48,57)(13,33,35,43,41,40,49,58)(24,45,50,54,53,62,52,51)(37,55,56,61,60,59,63,64)Polycyclic presentation:
2^8|a^2=d,b^2=c,c^2=h,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bc,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #94605, #94606, #94737, … |
| maximal normal subgroup of | #94605, #94606, #94737, … |
| maximal quotient of | #94745, #94747, #94757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 532);