Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,15,20,33)(4,21,25,40)(5,26,29,38)(6,13,7,12)(9,19,34,18)(10,30,37,24)(11,28,39,27)(16,35,22,31)(17,45,48,32)(23,53,55,36)(41,49,50,44)(42,47,58,46)(43,62,52,61)(51,56,60,54)(57,63,59,64);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,24,49,25,16,30,44)(5,17,28,47,29,48,27,46)(10,35,21,50,37,31,40,41)(11,32,38,58,39,45,26,42)(23,52,54,64,55,43,56,63)(36,59,51,62,53,57,60,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31)(11,36)(12,21)(13,40)(14,37)(15,41)(17,43)(18,44)(19,49)(20,22)(26,51)(27,54)(28,56)(29,55)(32,57)(33,50)(34,35)(38,60)(39,53)(42,61)(45,59)(46,63)(47,64)(48,52)(58,62);(1,5,7,28,8,29,6,27)(2,11,13,38,14,39,12,26)(3,17,19,47,20,48,18,46)(4,23,30,56,25,55,24,54)(9,32,33,58,34,45,15,42)(10,36,40,60,37,53,21,51)(16,43,49,64,22,52,44,63)(31,57,50,62,35,59,41,61)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501860, #501861, #501866, … |
| maximal normal subgroup of | #501860, #501861, #501866, … |
| maximal quotient of | #280845, #280847, #280851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17977);