Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
642 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 133 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:
(1,2,8,30,7,12,6,13)(3,14,28,33,18,9,17,39)(4,19,29,10,22,40,23,36)(5,24,26,37,27,11,25,38)(15,43,49,41,45,34,20,31)(16,32,47,58,48,42,46,57)(21,50,54,55,52,60,53,35)(44,62,51,56,63,59,64,61);(1,3,7,18)(2,9,12,14)(4,20,22,49)(5,16,27,48)(6,28,8,17)(10,34,36,43)(11,32,24,42)(13,39,30,33)(15,29,45,23)(19,31,40,41)(21,51,52,64)(25,47,26,46)(35,59,55,62)(37,58,38,57)(44,53,63,54)(50,56,60,61);(1,4)(2,10)(3,15)(5,21)(6,29)(7,22)(8,23)(9,31)(11,35)(12,36)(13,40)(14,41)(16,44)(17,49)(18,45)(19,30)(20,28)(24,55)(25,54)(26,53)(27,52)(32,56)(33,43)(34,39)(37,50)(38,60)(42,61)(46,51)(47,64)(48,63)(57,59)(58,62);(1,5)(2,11)(3,16)(4,21)(6,26)(7,27)(8,25)(9,32)(10,35)(12,24)(13,38)(14,42)(15,44)(17,47)(18,48)(19,50)(20,51)(22,52)(23,54)(28,46)(29,53)(30,37)(31,56)(33,58)(34,59)(36,55)(39,57)(40,60)(41,61)(43,62)(45,63)(49,64)Polycyclic presentation:
2^8|a^2=fh,b^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501146, #501149, #501150, … |
| maximal normal subgroup of | #501146, #501149, #501150, … |
| maximal quotient of | #267470, #267474, #267477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17411);