Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | Aut(G) = #72338915 (order 4096) · Inn(G) = #2668 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 427 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)(3,15,20,34)(4,21,26,38)(5,27,30,11)(6,13,7,12)(9,19,35,18)(10,24,39,25)(16,32,46,49)(17,40,47,43)(22,51,45,36)(23,50,55,61)(28,33,48,59)(29,41,31,42)(37,54,56,53)(44,58,63,64)(52,62,57,60);(1,3,8,20)(2,9,14,35)(4,22,26,45)(5,28,30,48)(6,19,7,18)(10,36,39,51)(11,40,27,43)(12,34,13,15)(16,25,46,24)(17,31,47,29)(21,49,38,32)(23,44,55,63)(33,42,59,41)(37,58,56,64)(50,60,61,62)(52,54,57,53);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,38)(13,21)(14,39)(15,36)(17,44)(18,22)(19,45)(20,46)(27,56)(28,57)(29,53)(30,55)(31,54)(33,58)(34,51)(35,49)(40,62)(41,50)(42,61)(43,60)(47,63)(48,52)(59,64);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,37)(12,42)(13,41)(14,27)(15,43)(16,44)(18,48)(19,28)(20,47)(21,50)(22,52)(24,54)(25,53)(26,55)(32,58)(34,40)(35,59)(36,60)(38,61)(39,56)(45,57)(46,63)(49,64)(51,62)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151319, #151321, #151323, … |
| maximal normal subgroup of | #151319, #151321, #151323, … |
| maximal quotient of | #331953, #331954, #331955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23431);