Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = C22×#72224639 (order 4096) · Inn(G) = D32 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D32 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(6,12)(7,17)(8,19)(9,24)(10,28)(13,35)(14,39)(15,42)(16,34)(18,23)(20,50)(21,54)(22,31)(25,59)(26,36)(27,32)(29,40)(30,33)(37,67)(38,66)(41,68)(43,44)(45,52)(46,60)(47,58)(48,49)(51,56)(53,64)(55,57)(61,65)(62,63);(5,6,11,16)(7,18,22,47)(8,23,26,58)(9,27,29,61)(10,30,28,15)(12,31,34,17)(13,36,38,19)(14,40,41,24)(20,51,53,67)(21,55,52,46)(25,60,48,57)(32,64,65,50)(33,45,42,54)(35,62,66,43)(37,49,56,59)(39,63,68,44)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ce,e^b=ef,f^b=fg,g^b=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7538, #153555, #153556, … |
| maximal normal subgroup of | C2×#7538, #153555, #153556, … |
| maximal quotient of | #94608, #94691, #94735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 526);