Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13605 |
| automorphism | Aut(G) = C2×#493397 (order 1024) · Inn(G) = #2127 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 43 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)(3,13,16,40)(4,9)(5,21,23,55)(6,25,27,11)(7,12)(8,15,29,46)(10,24)(14,39,43,61)(17,36,26,41)(18,49,22,54)(19,51,53,34)(20,35)(28,45,59,64)(30,47,31,48)(32,42,60,63)(33,50)(37,56)(38,57)(44,58,52,62);(1,3,7,17)(2,8,12,31)(4,14,20,44)(5,15,23,48)(6,26,27,16)(9,32,35,28)(10,36,38,13)(11,29,25,30)(18,42,22,45)(19,52,53,43)(21,41,55,40)(24,46,57,47)(33,58,37,39)(34,60,51,59)(49,62,54,61)(50,63,56,64);(1,4,7,20)(2,9,12,35)(3,14,17,44)(5,22,23,18)(6,19,27,53)(8,28,31,32)(10,37,38,33)(11,34,25,51)(13,39,36,58)(15,45,48,42)(16,43,26,52)(21,54,55,49)(24,56,57,50)(29,59,30,60)(40,61,41,62)(46,64,47,63)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,e^d=egh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #152140, #152141, #152142, … |
| maximal normal subgroup of | #152140, #152141, #152142, … |
| maximal quotient of | #149594, #149636, #149714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6596);