Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7548 |
| automorphism | Aut(G) = #72286351 (order 2048) · Inn(G) = #1424 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1424 |
| 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 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(2,8,10,26)(3,14)(4,19)(5,20)(6,23)(9,30)(11,39,36,53)(12,43,37,63)(13,44,38,29)(15,22)(16,33)(17,35)(18,34)(21,48)(24,51)(25,52,41,27)(28,56,55,40)(31,58,59,54)(32,61)(42,64,62,60)(45,46)(47,57)(49,50);(1,2,9,31,4,11,32,55,20,40,47,53,50,58,18,26,6,13,34,52,21,42,57,63,24,37,61,64,15,25,30,44,7,10,35,59,22,36,16,28,51,56,46,39,48,54,3,8,23,38,14,27,49,62,45,43,5,12,33,60,19,41,17,29)Polycyclic presentation:
2^8|a^2=d,b^2=ceg,c^2=efh,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7548, #153578, #153597, … |
| maximal normal subgroup of | C2×#7548, #153578, #153597, … |
| maximal quotient of | #94752, #94764, C2×#7548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 536);