Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1964 |
| commutator | C22×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,7,12,8,13,6,28)(3,14,18,41,19,42,29,40)(4,20,23,37,24,51,22,10)(5,25,15,38,27,11,30,32)(9,31,34,26,35,48,58,17)(16,44,46,39,47,61,55,62)(21,52,43,57,54,50,56,36)(33,45,49,59,60,53,64,63);(1,3,15,31)(2,9,32,41)(4,16,43,59)(5,26,7,29)(6,18,27,17)(8,19,30,48)(10,33,52,61)(11,14,12,34)(13,35,38,40)(20,49,36,39)(21,53,23,55)(22,46,54,45)(24,47,56,63)(25,42,28,58)(37,60,50,44)(51,64,57,62);(1,4,7,23,8,24,6,22)(2,10,28,51,13,37,12,20)(3,16,29,55,19,47,18,46)(5,21,30,56,27,54,15,43)(9,33,58,64,35,60,34,49)(11,36,32,52,25,57,38,50)(14,39,41,61,42,62,40,44)(17,45,31,59,26,53,48,63)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fg,g^b=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #138936, #138937, #138950, … |
| maximal normal subgroup of | #138936, #138937, #138950, … |
| maximal quotient of | #140616, #140634, #140646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6221);