Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | Aut(G) = #72305500 (order 2048) · Inn(G) = #1505 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1505 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 146 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)(3,14,17,43)(4,10)(5,25,27,41)(6,12)(7,30)(8,13)(9,16)(11,28,39,29)(15,42,45,60)(18,38,19,49)(20,50,46,64)(21,52,53,62)(22,36)(23,56)(24,37)(26,40)(31,44)(32,47)(33,48)(34,51)(35,54,61,55)(57,59)(58,63);(1,3,8,17)(2,9,13,33)(4,20,24,46)(5,26,27,47)(6,19,7,18)(10,34,37,59)(11,38,39,49)(12,40,30,32)(14,41,43,25)(15,23,45,22)(16,28,48,29)(21,44,53,63)(31,56,58,36)(35,60,61,42)(50,62,64,52)(51,54,57,55);(1,4,7,23,8,24,6,22)(2,10,30,56,13,37,12,36)(3,15,19,46,17,45,18,20)(5,21,29,55,27,53,28,54)(9,31,40,59,33,58,32,34)(11,35,41,62,39,61,25,52)(14,42,49,64,43,60,38,50)(16,44,26,57,48,63,47,51)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=fh,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #141373, #141375, #141376, … |
| maximal normal subgroup of | #141373, #141375, #141376, … |
| maximal quotient of | #124371, #124462, #124502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4759);