Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×C8×AGL(1,ℤ/8) (order 512) · Inn(G) = D4 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | D4×C16 |
| prime radical | #135 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,13,28,42,5,14,25,43,26,45,56,63,8,15,29,44,30,46,58,64,27,11,38,40,57,41,32,59)(3,16,18,47,19,22,36,53,20,34,49,60,51,24,37,55,21,39,50,62,52,54,35,61,17,9,31,33,48,4,10,23);(1,3)(2,9)(4,13)(5,20)(6,18)(7,19)(8,21)(10,32)(11,39)(12,33)(14,16)(15,34)(17,27)(22,45)(23,42)(24,46)(25,49)(26,51)(28,36)(29,50)(30,52)(31,38)(35,58)(37,56)(40,62)(41,54)(43,47)(44,60)(48,57)(53,63)(55,64)(59,61);(1,4,6,23,7,16,28,47,5,22,25,53,26,34,56,60,8,24,29,55,30,39,58,62,27,54,38,61,57,9,32,33)(2,10,12,3,13,18,42,19,14,36,43,20,45,49,63,51,15,37,44,21,46,50,64,52,11,35,40,17,41,31,59,48)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=h,e^2=f,f^2=g,g^2=h,b^a=bd,d^a=dh,d^b=dh112 conjugacy classes — view the full character table.
| maximal subgroup of | #94605, #94606, #151072, … |
| maximal normal subgroup of | #94605, #94606, #151072, … |
| maximal quotient of | D8×C32, #151071, SD16×C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6635);