Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | Aut(G) = #72287443 (order 4096) · Inn(G) = #1415 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1415 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| prime radical | C22 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,11,8,15,32,14)(3,16,18,62,21,66,29,65)(4,22,25,78,28,82,90,81)(6,33,30,98,36,48,100,101)(7,37,31,13,39,54,76,60)(9,40,41,17,45,68,44,71)(10,46,49,92,52,111,51,26)(12,55,53,24,58,87,57,35)(19,73,72,118,74,114,96,124)(20,75,38,64,77,113,97,122)(23,56,83,121,69,61,88,120)(27,93,89,80,95,126,125,116)(34,63,103,115,67,47,99,112)(42,94,105,102,108,128,107,85)(43,70,106,104,109,123,59,84)(50,86,117,127,119,91,79,110);(1,3,8,21)(2,9,15,45)(4,23,28,69)(5,29,32,18)(6,34,36,67)(7,38,39,97)(10,47,52,63)(11,44,14,41)(12,56,58,61)(13,59,60,106)(16,22,66,82)(17,26,71,92)(19,24,74,35)(20,76,77,31)(25,88,90,83)(27,94,95,128)(30,99,100,103)(33,42,48,108)(37,43,54,109)(40,46,68,111)(49,115,51,112)(50,118,119,124)(53,120,57,121)(55,96,87,72)(62,81,65,78)(64,93,122,126)(70,110,123,127)(73,79,114,117)(75,80,113,116)(84,86,104,91)(85,125,102,89)(98,107,101,105)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=efh,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=ceh,d^b=dh,e^b=ef,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7542, #153565, #153597, … |
| maximal normal subgroup of | C2×#7542, #153565, #153597, … |
| maximal quotient of | #94610, #94693, #94735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 530);