Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | Aut(G) = #72305500 (order 2048) · Inn(G) = #1505 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1505 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 47 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)(3,8)(4,17,19,7)(5,10)(6,12)(9,18,21,28)(11,14)(13,23)(15,29,30,20)(16,25)(22,26,31,32)(24,27);(2,7)(3,13)(4,18)(5,14)(8,22)(9,11)(10,21)(12,17)(15,30)(16,27)(19,28)(20,23)(24,29)(25,31);(1,3,14,27,6,16,5,13)(2,8,11,24,12,25,10,23)(4,15,28,32,19,30,18,26)(7,20,21,31,17,29,9,22)Polycyclic presentation:
2^8|a^2=e,c^2=fh,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #141373, #141374, #141376, … |
| maximal normal subgroup of | #141373, #141374, #141376, … |
| maximal quotient of | #124370, #124461, #124502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4758);