Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = #353 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #407 | C22 |
| socle | C2 | #1343 |
| prime radical | #2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
134 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 21 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 — 2 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,15,29,17,30,16,26)(4,18,20,27,21,28,22,9)(8,19,23,14,25,31,24,32);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,24)(11,23)(12,25)(13,18)(14,20)(21,31)(22,32)(27,29)(28,30)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gh,d^c=dgh,e^c=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7397, #150173, #150326, … |
| maximal normal subgroup of | C2×#7397, #150173, #150326, … |
| maximal quotient of | #93225, #93226, #93227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 385);