Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,25)(21,31)(22,28)(23,27)(24,29)(30,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,21,24,30,22,20)(18,25,28,32,29,31,27,26);(11,15)(13,16)(17,20,23,19,24,21,22,30)(18,26,28,25,29,32,27,31);(1,3,4,8)(2,6,5,7)(17,21,23,30,24,20,22,19)(18,26,28,25,29,32,27,31)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=f,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #504630, #504752, #505268, … |
| maximal normal subgroup of | #504630, #504752, #505268, … |
| maximal quotient of | #360933, #362546, #367559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20991);