Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1898 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1898 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,13,18,42)(4,19,22,38)(5,23,14,10)(6,11)(7,12)(8,16)(9,21,34,28)(15,44)(17,41,27,58)(20,53)(24,32)(25,52)(26,55,43,36)(29,39)(30,59,62,64)(31,49)(33,54)(35,57)(37,56)(40,50)(45,51,48,61)(46,60)(47,63);(1,3,14,16,7,18,5,24)(2,8,23,42,12,32,10,13)(4,15,20,48,22,47,25,45)(6,27,43,56,29,17,26,49)(9,30,33,60,34,62,35,40)(11,37,55,41,39,31,36,58)(19,51,52,44,38,61,53,63)(21,50,54,64,28,46,57,59);(1,4)(2,9)(3,15)(5,25)(6,28)(7,22)(8,30)(10,35)(11,38)(12,34)(13,40)(14,20)(16,48)(17,50)(18,47)(19,39)(21,29)(23,33)(24,45)(26,54)(27,46)(31,51)(32,62)(36,52)(37,61)(41,63)(42,60)(43,57)(44,58)(49,64)(53,55)(56,59)Polycyclic presentation:
2^8|a^2=f,b^2=df,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13096, #494748, #494749, … |
| maximal normal subgroup of | C2×#13096, #494748, #494749, … |
| maximal quotient of | #137684, #137691, #137731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6084);