Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #2021 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 3 generators on 64 points:
(1,2,7,12)(3,13,17,35)(4,18,21,37)(5,22,25,10)(6,26,28,11)(8,23,31,15)(9,27,34,20)(14,40,43,64)(16,46,48,39)(19,51,53,50)(24,36,58,55)(29,60,32,63)(30,45,59,56)(33,52,54,57)(38,47,61,42)(41,49,44,62);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,32)(10,35)(11,30)(12,31)(13,22)(15,25)(18,49)(19,44)(20,42)(21,43)(24,56)(26,59)(27,47)(28,48)(29,34)(33,61)(36,46)(37,62)(38,54)(39,55)(40,51)(41,53)(45,58)(50,64)(52,63)(57,60);(1,4,7,21)(2,9,12,34)(3,14,17,43)(5,19,25,53)(6,27,28,20)(8,29,31,32)(10,33,22,54)(11,37,26,18)(13,38,35,61)(15,44,23,41)(16,47,48,42)(24,57,58,52)(30,49,59,62)(36,51,55,50)(39,64,46,40)(45,60,56,63)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,g^b=gh,d^c=dg,e^c=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13356, #498940, #498942, … |
| maximal normal subgroup of | C2×#13356, #498940, #498942, … |
| maximal quotient of | #142227, #142231, #142274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6344);