Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = C4×#72218857 (order 6144) · Inn(G) = #120 · Out(G) = C2×C4×S4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #120 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1779 |
| prime radical | #585 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,6,11,20,30,7,12,21,31,22,32,43,53)(3,8,15,25,16,26,39,49,14,24,37,47,38,48,58,62)(4,9,18,28,19,29,42,52,17,27,40,50,41,51,59,63)(13,33,35,54,36,55,57,64,34,23,44,45,56,46,60,61);(1,3)(2,8)(4,13)(5,15)(6,16)(7,14)(9,23)(10,25)(11,26)(12,24)(17,34)(18,35)(19,36)(20,39)(21,37)(22,38)(27,33)(28,45)(29,46)(30,49)(31,47)(32,48)(40,44)(41,56)(42,57)(43,58)(50,54)(51,55)(52,61)(53,62)(59,60)(63,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,23)(10,28)(11,29)(12,27)(14,34)(15,35)(16,36)(20,42)(21,40)(22,41)(24,33)(25,45)(26,46)(30,52)(31,50)(32,51)(37,44)(38,56)(39,57)(43,59)(47,54)(48,55)(49,61)(53,63)(58,60)(62,64)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=bd,c^a=ce,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #12684⋊C3 |
| maximal subgroup of | #136829, #136830, #136831, … |
| maximal normal subgroup of | #136829, #136830, #136831, … |
| maximal quotient of | #129204, #129243, #129280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5672);