Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/32) (order 4096) · Inn(G) = #2215 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,9,16)(3,17,24,60)(4,25,29,52)(5,30,34,89)(6,15,8,13)(7,37,38,53)(10,39,44,20)(11,35,47,27)(12,48,51,18)(14,36,54,31)(19,65,69,109)(21,59,23,57)(22,75,76,110)(26,80,83,123)(28,84,85,46)(32,88,33,87)(40,94,96,118)(41,71,43,72)(42,99,100,113)(45,101,102,114)(49,63,50,64)(55,73,108,67)(56,93,98,61)(58,74,111,70)(62,78,103,104)(66,77,92,97)(68,119,120,107)(79,91,105,81)(82,86,90,122)(95,112,115,124)(106,125,126,128)(116,121,127,117);(1,3,18,61,7,22,32,71,6,21,63,99,36,74,34,39,9,24,48,93,38,76,33,72,8,23,64,113,31,70,5,20)(2,10,30,58,14,42,49,57,13,41,87,110,53,98,51,60,16,44,89,111,54,100,50,59,15,43,88,75,37,56,12,17)(4,19,62,112,28,68,90,94,35,73,101,125,91,121,83,92,29,69,103,124,85,120,82,118,27,67,114,128,81,117,26,66)(11,40,86,107,46,95,104,109,52,97,123,127,105,126,102,108,47,96,122,119,84,115,78,65,25,77,80,116,79,106,45,55);(1,4)(2,11)(3,19)(5,26)(6,35)(7,28)(8,27)(9,29)(10,40)(12,45)(13,52)(14,46)(15,25)(16,47)(17,55)(18,62)(20,66)(21,73)(22,68)(23,67)(24,69)(30,86)(31,81)(32,90)(33,82)(34,83)(36,91)(37,79)(38,85)(39,92)(41,97)(42,95)(43,77)(44,96)(48,103)(49,104)(50,78)(51,102)(53,105)(54,84)(56,106)(57,109)(58,107)(59,65)(60,108)(61,112)(63,101)(64,114)(70,117)(71,94)(72,118)(74,121)(75,116)(76,120)(80,88)(87,123)(89,122)(93,124)(98,126)(99,125)(100,115)(110,127)(111,119)(113,128)Polycyclic presentation:
2^8|a^2=h,b^2=df,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #153611, #153612, #153665, … |
| maximal normal subgroup of | #153611, #153612, #153665, … |
| maximal quotient of | #151638, #151641, #151647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6689);