Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
225 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,19,12,15)(13,18)(14,17)(16,21,23,20)(22,24);(1,2,3,7,4,6,8,5)(9,11,13,12)(10,15)(14,21,22,20)(16,17)(18,19)(23,24);(9,12,13,11)(10,16,18,23)(14,20,22,21)(15,17,19,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9160, #196988, #196989, … |
| maximal normal subgroup of | C2×#9160, #196988, #196989, … |
| maximal quotient of | #99788, #99789, #99796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2148);