Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = C2×#72224639 (order 2048) · Inn(G) = D32 · Out(G) = C2×C16 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D32 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| socle | C2 | #1415 |
| prime radical | #2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,9,15,8,40)(3,16,18,59,21,62,41,61)(4,22,25,70,29,73,28,88)(6,32,30,90,36,93,35,96)(7,37,31,58,39,98,34,14)(10,42,43,64,47,101,46,17)(11,48,50,84,52,83,100,26)(13,54,53,80,56,79,92,24)(19,66,65,120,67,121,94,112)(20,68,38,60,69,117,99,118)(23,55,74,114,77,116,89,115)(27,85,81,123,87,127,82,72)(33,49,91,109,95,111,97,110)(44,86,102,125,105,128,104,76)(45,106,103,124,107,63,57,75)(51,78,71,108,113,126,122,119);(1,3)(2,10)(4,23)(5,18)(6,33)(7,38)(8,41)(9,21)(11,49)(12,43)(13,55)(14,57)(15,47)(16,22)(17,26)(19,24)(20,34)(25,74)(27,86)(28,89)(29,77)(30,91)(31,69)(32,44)(35,97)(36,95)(37,45)(39,99)(40,46)(42,48)(50,109)(51,112)(52,111)(53,114)(54,65)(56,116)(58,103)(59,70)(60,85)(61,88)(62,73)(63,119)(64,84)(66,71)(67,80)(68,72)(75,78)(76,82)(79,94)(81,125)(83,101)(87,128)(90,102)(92,115)(93,105)(96,104)(98,107)(100,110)(106,108)(113,120)(117,123)(118,127)(121,122)(124,126)Polycyclic presentation:
2^8|a^2=d,c^2=ef,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ce,e^b=ef,f^b=fg,g^b=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7539, #153560, #153595, … |
| maximal normal subgroup of | C2×#7539, #153560, #153595, … |
| maximal quotient of | #94607, #94608, #94692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 527);