Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/32) (order 12288) · Inn(G) = C22×D16 · Out(G) = C2×D3×C8 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D16 |
| commutator, Frattini, cosocle | C16 | C24 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,14)(3,15,20,57)(4,10,21,42)(5,25,30,54)(6,31,34,87)(7,35,33,13)(9,37,41,17)(11,46,49,28)(12,50,52,27)(16,55,58,107)(18,63,64,104)(19,65,36,56)(22,69,66,106)(23,74,67,115)(24,76,68,45)(26,51,82,105)(29,85,84,79)(32,47,88,103)(38,89,90,59)(39,86,93,81)(40,62,53,80)(43,96,94,72)(44,99,95,71)(48,83,78,102)(60,111,109,128)(61,112,77,108)(70,100,113,126)(73,122,114,118)(75,97,116,125)(91,123,124,120)(92,110,101,119)(98,121,117,127);(1,3,8,20)(2,9,14,41)(4,16,21,58)(5,26,30,82)(6,32,34,88)(7,36,33,19)(10,38,42,90)(11,47,49,103)(12,51,52,105)(13,53,35,40)(15,54,57,25)(17,46,37,28)(18,50,64,27)(22,70,66,113)(23,75,67,116)(24,77,68,61)(29,86,84,81)(31,39,87,93)(43,97,94,125)(44,100,95,126)(45,101,76,92)(48,104,78,63)(55,106,107,69)(56,79,65,85)(59,96,89,72)(60,99,109,71)(62,83,80,102)(73,123,114,120)(74,91,115,124)(98,128,117,111)(108,118,112,122)(110,121,119,127);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,21)(9,38)(11,43)(12,44)(13,45)(14,42)(15,55)(17,59)(18,60)(19,61)(20,58)(25,69)(26,70)(27,71)(28,72)(29,73)(30,66)(31,74)(32,75)(33,68)(34,67)(35,76)(36,77)(37,89)(39,91)(40,92)(41,90)(46,96)(47,97)(48,98)(49,94)(50,99)(51,100)(52,95)(53,101)(54,106)(56,108)(57,107)(62,110)(63,111)(64,109)(65,112)(78,117)(79,118)(80,119)(81,120)(82,113)(83,121)(84,114)(85,122)(86,123)(87,115)(88,116)(93,124)(102,127)(103,125)(104,128)(105,126);(4,21)(10,42)(16,58)(22,66)(23,67)(24,68)(38,90)(43,94)(44,95)(45,76)(55,107)(59,89)(60,109)(61,77)(69,106)(70,113)(71,99)(72,96)(73,114)(74,115)(75,116)(91,124)(92,101)(97,125)(98,117)(100,126)(108,112)(110,119)(111,128)(118,122)(120,123)(121,127)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=fg,f^2=gh,g^2=h,b^a=be,e^a=ef,f^a=fg,g^a=gh,e^b=ef,f^b=fg,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #33984⋊C3 |
| maximal subgroup of | #153610, #153612, #513189, … |
| maximal normal subgroup of | #153610, #153612, #513189, … |
| maximal quotient of | #512692, #512719, #512765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26972);