Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13578 |
| automorphism | Aut(G) = #72317530 (order 2048) · Inn(G) = C2×#353 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 210 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)(3,11,14,26)(4,15,16,31)(5,18,20,9)(6,10)(7,13,22,29)(8,17)(12,24,19,27)(21,28,23,30)(25,32);(1,3)(2,7)(4,13)(5,19)(6,12)(8,24)(9,22)(10,21)(11,25)(14,20)(15,27)(16,28)(17,29)(18,23)(26,31)(30,32);(3,12)(7,21)(11,24)(13,28)(14,19)(22,23)(26,27)(29,30)Polycyclic presentation:
2^8|a^2=e,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #152047, #152048, #152082, … |
| maximal normal subgroup of | #152047, #152048, #152082, … |
| maximal quotient of | #149568, #149629, #149687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6566);