Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | Aut(G) order 4096 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 539 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)(3,15)(4,20)(5,24)(6,14)(7,13)(8,11)(9,17)(10,22)(12,27)(16,43)(18,49)(19,42)(21,52)(23,39)(25,40)(26,36)(28,38)(29,34)(30,37)(31,58)(32,48)(33,56)(35,55)(41,45)(44,50)(46,63)(47,61)(51,54)(53,59)(57,62)(60,64);(1,3,8,19)(2,9,11,32)(4,16,23,47)(5,25,28,56)(6,18,7,30)(10,31,34,59)(12,37,36,49)(13,33,14,40)(15,38,42,24)(17,26,48,27)(20,50,39,60)(21,53,55,58)(22,45,29,46)(35,61,52,43)(41,62,63,51)(44,57,64,54);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,31)(11,34)(12,35)(13,39)(14,20)(15,41)(17,44)(18,46)(19,47)(24,51)(25,53)(26,57)(27,54)(28,55)(30,45)(32,59)(33,60)(36,52)(37,61)(38,62)(40,50)(42,63)(43,49)(48,64)(56,58);(2,11)(9,32)(10,34)(12,36)(13,14)(15,42)(20,39)(24,38)(31,59)(33,40)(35,52)(37,49)(41,63)(43,61)(50,60)(51,62)Polycyclic presentation:
2^8|b^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gh,e^b=eg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #495740, #498125, #499748, … |
| maximal normal subgroup of | #495740, #498125, #499748, … |
| maximal quotient of | #484213, #484214, #484222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26528);