Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = C4×#511819 (order 2048) · Inn(G) = #117 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1746 |
| prime radical | C23 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,6,12,22,34,7,11,23,33,24,35,49,59)(3,13,16,37,17,38,42,61,15,8,25,27,41,28,51,54)(4,9,19,30,21,31,47,57,20,32,46,58,48,52,63,64)(14,36,40,60,18,43,45,62,39,26,50,53,44,29,55,56);(1,3)(2,8,11,13)(4,18,20,44)(5,16)(6,17)(7,15)(9,29)(10,27,33,37)(12,28,35,38)(14,21,39,48)(19,45,46,55)(22,42)(23,25)(24,41)(26,52)(30,56)(31,36)(32,43)(34,54,59,61)(40,47,50,63)(49,51)(53,64)(57,60)(58,62);(1,4)(2,9)(3,14,15,39)(5,19)(6,21)(7,20)(8,26,13,36)(10,30)(11,32)(12,31)(16,40,25,50)(17,18,41,44)(22,47)(23,46)(24,48)(27,53,37,60)(28,29,38,43)(33,58)(34,57)(35,52)(42,45,51,55)(49,63)(54,56,61,62)(59,64)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,e^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94003, #94004, #141376, … |
| maximal normal subgroup of | #94003, #94004, #141376, … |
| maximal quotient of | #128213, #128218, #128221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5585);