Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#466 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
318 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,18,41)(4,22,25,39)(5,26,28,37)(6,13,7,12)(9,30,32,19)(10,29,36,24)(11,27,38,16)(17,46,47,63)(20,43,21,42)(23,54,55,35)(31,58,59,64)(33,50,34,44)(40,51,62,49)(45,53,56,61)(48,52,57,60);(1,3,16,44,7,21,28,30,8,18,27,50,6,20,5,19)(2,9,26,43,13,34,38,41,14,32,37,42,12,33,11,15)(4,17,45,64,24,49,55,57,25,47,56,58,29,51,23,48)(10,31,53,46,22,52,54,62,36,59,61,63,39,60,35,40);(1,4)(2,10)(3,17)(5,23)(6,29)(7,24)(8,25)(9,31)(11,35)(12,39)(13,22)(14,36)(15,40)(16,45)(18,47)(19,48)(20,51)(21,49)(26,53)(27,56)(28,55)(30,57)(32,59)(33,60)(34,52)(37,61)(38,54)(41,62)(42,63)(43,46)(44,64)(50,58);(3,18)(9,32)(15,41)(17,47)(19,30)(20,21)(31,59)(33,34)(40,62)(42,43)(44,50)(46,63)(48,57)(49,51)(52,60)(58,64)Polycyclic presentation:
2^8|a^2=h,b^2=eg,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gh,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #495900, #498380, #501063, … |
| maximal normal subgroup of | #495900, #498380, #501063, … |
| maximal quotient of | #490473, #490480, #490543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26607);