Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C8 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1316 |
| prime radical | #585 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,26,44,8,13,27,45,6,28,31,43)(3,14,16,46,17,47,51,63,18,38,52,60,29,59,24,56)(4,19,21,53,22,10,37,39,23,40,25,57,30,41,55,64)(9,32,33,15,35,48,42,50,36,58,61,20,34,54,62,49);(1,3)(2,9)(4,20)(5,24)(6,29)(7,17)(8,18)(10,38)(11,42)(12,35)(13,36)(14,41)(15,23)(16,26)(19,47)(21,32)(22,49)(25,58)(27,51)(28,34)(30,50)(31,52)(33,43)(37,48)(39,63)(40,59)(44,61)(45,62)(46,53)(54,55)(56,64)(57,60)Polycyclic presentation:
2^8|a^2=d,c^2=h,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=de,e^b=eh,g^b=gh,d^c=dgh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7347, #149548, #149683, … |
| maximal normal subgroup of | C2×#7347, #149548, #149683, … |
| maximal quotient of | #93196, #93197, #93198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 335);