Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | Aut(G) = #72230720 (order 4096) · Inn(G) = D64 · Out(G) = C32 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| semi dihedral | SD256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D64 |
| commutator, Frattini, cosocle | C64 | C22 |
| prime radical | D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ G | |
| LECS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(2,64)(3,127)(4,62)(5,125)(6,60)(7,123)(8,58)(9,121)(10,56)(11,119)(12,54)(13,117)(14,52)(15,115)(16,50)(17,113)(18,48)(19,111)(20,46)(21,109)(22,44)(23,107)(24,42)(25,105)(26,40)(27,103)(28,38)(29,101)(30,36)(31,99)(32,34)(33,97)(35,95)(37,93)(39,91)(41,89)(43,87)(45,85)(47,83)(49,81)(51,79)(53,77)(55,75)(57,73)(59,71)(61,69)(63,67)(66,128)(68,126)(70,124)(72,122)(74,120)(76,118)(78,116)(80,114)(82,112)(84,110)(86,108)(88,106)(90,104)(92,102)(94,100)(96,98)Polycyclic presentation:
2^8|a^2=h,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh67 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD256, #153705, #153706, … |
| maximal normal subgroup of | C2×SD256, #153705, #153706, … |
| maximal quotient of | #94832, #94833, #94841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 540);