Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) = #72287717 (order 8192) · Inn(G) = C2×D16 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,28)(10,32)(11,13)(12,18)(14,25)(15,24)(16,30)(19,49)(20,52)(21,44)(23,41)(26,60)(27,43)(29,39)(31,62)(33,45)(34,48)(35,56)(36,55)(37,46)(38,50)(40,67)(42,47)(51,68)(53,58)(54,59)(57,66)(61,64)(63,65);(5,7,11,21)(6,12,13,34)(8,23,27,57)(9,29,31,63)(10,33,30,20)(14,39,37,65)(15,41,38,66)(16,42,32,36)(17,43,44,22)(18,46,48,25)(19,50,51,24)(26,61,59,56)(28,35,62,64)(40,68,53,49)(45,54,52,60)(47,58,55,67);(1,3,4,2)(6,13)(12,34)(14,37)(15,38)(16,32)(17,44)(22,43)(28,62)(35,64)(36,42)(39,65)(40,53)(41,66)(45,52)(49,68)(54,60)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=e,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94615, #151084, #151118, … |
| maximal normal subgroup of | #94615, #151084, #151118, … |
| maximal quotient of | #151079, #151114, #151125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6641);