Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72286320 (order 2048) · Inn(G) = #354 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| socle | C2 | C2×#354 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13,16,23)(4,9)(5,20)(6,11,22,26)(7,12)(8,15,24,32)(10,21,28,19)(14,29,31,25)(17,30,18,27);(1,3,7,18)(2,8)(4,14,5,15)(6,16,22,17)(9,23)(10,27)(11,29)(12,25)(13,28)(19,31,21,32)(20,30)(24,26);(1,4,7,5)(2,9,12,20)(3,14,18,15)(6,19,22,21)(8,23,25,30)(10,26,28,11)(13,29,27,24)(16,31,17,32)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dgh,e^b=efgh,f^b=fh,e^d=egh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151941, #151942, #151945, … |
| maximal normal subgroup of | #151941, #151942, #151945, … |
| maximal quotient of | #149586, #149610, #149705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6590);