Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2,7,12)(3,13,16,42)(4,9,18,33)(5,21,26,56)(6,27,24,11)(8,29,31,15)(10,36,39,25)(14,43,44,41)(17,48,28,37)(19,50,23,55)(20,53,49,35)(22,40,58,32)(30,47,60,45)(34,59,38,52)(46,64,54,62)(51,63,57,61);(1,3)(2,8)(4,14)(5,22)(6,28)(7,16)(9,30)(10,37)(11,40)(12,31)(13,21)(15,25)(17,24)(18,44)(19,51)(20,54)(23,57)(26,58)(27,32)(29,36)(33,60)(34,62)(35,63)(38,64)(39,48)(41,55)(42,56)(43,50)(45,52)(46,49)(47,59)(53,61);(1,4)(2,9)(3,14)(5,23)(6,20)(7,18)(8,30)(10,38)(11,35)(12,33)(13,41)(15,47)(16,44)(17,46)(19,26)(21,55)(22,57)(24,49)(25,59)(27,53)(28,54)(29,45)(31,60)(32,61)(34,39)(36,52)(37,64)(40,63)(42,43)(48,62)(50,56)(51,58);(4,18)(9,33)(14,44)(19,23)(20,49)(30,60)(34,38)(35,53)(41,43)(45,47)(46,54)(50,55)(51,57)(52,59)(61,63)(62,64)Polycyclic presentation:
2^8|a^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,f^b=fh,g^b=gh,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142618, #142619, #142622, … |
| maximal normal subgroup of | #142618, #142619, #142622, … |
| maximal quotient of | #484200, #484202, #484545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26520);