Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 544 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)(5,11)(6,26)(7,13)(8,14)(9,18)(10,23)(12,28)(16,44)(17,42)(19,43)(21,36)(22,53)(24,40)(25,39)(27,32)(29,38)(30,37)(31,58)(33,57)(34,49)(35,56)(41,47)(45,63)(46,50)(48,61)(51,55)(52,62)(54,60)(59,64);(1,3)(2,9)(4,16)(5,17)(6,27)(7,30)(8,19)(10,31)(11,32)(12,37)(13,33)(14,34)(15,26)(18,28)(20,50)(21,45)(22,54)(23,47)(24,48)(25,49)(29,57)(35,61)(36,60)(38,43)(39,42)(40,59)(41,51)(44,53)(46,55)(52,64)(56,58)(62,63);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,31)(11,20)(12,35)(13,40)(14,36)(15,41)(17,45)(18,46)(19,48)(25,52)(26,51)(27,54)(28,55)(29,56)(30,47)(32,50)(33,59)(34,60)(37,61)(38,62)(39,53)(42,44)(43,63)(49,64)(57,58);(1,5,8,7)(2,11,14,13)(3,17,19,30)(4,21,24,23)(6,25,29,28)(9,32,34,33)(10,20,36,40)(12,26,39,38)(15,42,43,37)(16,45,48,47)(18,27,49,57)(22,52,56,55)(31,50,60,59)(35,51,53,62)(41,44,63,61)(46,54,64,58)Polycyclic presentation:
2^8|d^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gh,e^b=eg,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #495719, #495720, #498172, … |
| maximal normal subgroup of | #495719, #495720, #498172, … |
| maximal quotient of | #484211, #484221, #484222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26526);