Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
923 conjugacy classes of subgroups; only the 403 normal subgroups are shown.
This group has 923 conjugacy classes of subgroups — too many to draw. Only its 403 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) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(18,23)(20,27)(25,29)(28,32);(2,5)(6,7)(10,14)(11,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130069, #133430, #133814, … |
| maximal normal subgroup of | #130069, #133430, #133814, … |
| maximal quotient of | #576606, #576614, #577215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39986);