Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,11,18,35,51,36,17,13,20,9,16,12,19,10,37)(7,21,25,44,39,42,24,14,27,43,23,53,26,50,38,45)(8,28,31,49,40,46,34,48,33,52,29,15,32,47,30,54)(22,56,59,55,63,68,58,64,61,67,57,65,60,41,62,66);(1,2,3,4)(5,7)(6,14)(8,22)(9,24)(10,25)(11,38)(12,39)(13,27)(15,41)(16,50)(17,21)(18,42)(19,53)(20,45)(23,36)(26,35)(28,64)(29,58)(30,59)(31,62)(32,63)(33,61)(34,57)(37,43)(40,60)(44,51)(46,55)(47,65)(48,56)(49,68)(52,66)(54,67);(5,8,13,33)(6,15,20,49)(7,22,27,61)(9,34,11,30)(10,31,36,29)(12,40,35,32)(14,41,45,68)(16,28,18,52)(17,46,37,47)(19,54,51,48)(21,55,43,65)(23,58,25,62)(24,57,38,59)(26,63,39,60)(42,66,50,64)(44,56,53,67)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13020, #494517, #494519, … |
| maximal normal subgroup of | C2×#13020, #494517, #494519, … |
| maximal quotient of | #136692, #136726, #136901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6008);