Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #466 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1893 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,14,19,39)(4,20,23,38)(5,24)(6,12)(7,31)(8,13)(9,17,34,25)(10,22,37,30)(11,15)(16,47)(18,43,29,44)(21,36,26,40)(27,41)(28,57)(32,58)(33,53,35,45)(42,50)(46,62,56,55)(48,60)(49,59)(51,63)(52,54)(61,64);(1,3,15,45,6,29,5,25,8,19,27,53,7,18,28,17)(2,9,24,43,31,33,11,39,13,34,57,44,12,35,41,14)(4,16,46,58,30,49,26,48,23,51,56,64,22,50,21,52)(10,32,55,63,20,54,40,59,37,61,62,47,38,60,36,42);(1,4)(2,10)(3,16)(5,26)(6,30)(7,22)(8,23)(9,32)(11,40)(12,38)(13,37)(14,42)(15,46)(17,52)(18,50)(19,51)(20,31)(21,28)(24,55)(25,48)(27,56)(29,49)(33,54)(34,61)(35,60)(36,41)(39,59)(43,63)(44,47)(45,58)(53,64)(57,62)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13071, #494655, #495893, … |
| maximal normal subgroup of | C2×#13071, #494655, #495893, … |
| maximal quotient of | #137617, #137618, #137657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6059);