Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13603 |
| automorphism | Aut(G) = #72312480 (order 2048) · Inn(G) = C2×#354 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
161 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 161 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)(3,13,15,26)(4,9)(5,20)(6,11,22,27)(7,12)(8,14,24,31)(10,21,29,19)(16,23,17,28)(18,30,32,25);(1,3,7,17)(2,8)(4,18,5,14)(6,15,22,16)(9,26)(10,28)(11,30)(12,25)(13,29)(19,32,21,31)(20,23)(24,27);(1,4,7,5)(2,9,12,20)(3,14,17,18)(6,19,22,21)(8,23,25,26)(10,27,29,11)(13,24,28,30)(15,31,16,32)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,c^b=ch,d^b=dgh,e^b=efgh,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #152073, #152074, #152089, … |
| maximal normal subgroup of | #152073, #152074, #152089, … |
| maximal quotient of | #149588, #149612, #149706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6592);