Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 363 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(9,11,12,16)(10,13,14,15)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,3,4,8)(2,6,5,7)(18,24)(19,27)(23,25)(28,32)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #150515, #150518, #150885, … |
| maximal normal subgroup of | #150515, #150518, #150885, … |
| maximal quotient of | #326586, #326593, #326602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22256);