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 | C2×D32 | C2 |
| 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 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,11,8,14,7,32)(3,15,18,42,22,9,21,33)(5,25,23,58,29,38,28,60)(6,30,24,46,31,61,27,13)(10,36,39,16,41,49,62,19)(12,44,43,17,37,52,59,26)(20,55,50,35,57,45,53,48)(34,63,40,54,47,56,64,51);(2,9)(3,16)(4,7)(5,26)(6,27)(10,37)(11,42)(12,41)(13,45)(14,15)(17,29)(18,36)(19,22)(20,56)(21,49)(23,52)(24,31)(25,34)(28,44)(30,35)(32,33)(38,47)(39,43)(40,60)(46,55)(48,61)(50,54)(51,53)(57,63)(58,64)(59,62)Polycyclic presentation:
2^8|a^2=d,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×#7541, #153565, #153598, … |
| maximal normal subgroup of | C2×#7541, #153565, #153598, … |
| maximal quotient of | #94609, #94692, #94734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 529);