Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 115 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)(17,18,22,25,21,24,29,32)(19,20,27,28,26,23,31,30);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,32)(29,31);(11,15)(13,16)(17,20,21,23)(18,19,24,26)(22,30,29,28)(25,31,32,27);(1,3,4,8)(2,6,5,7)(18,24)(19,26)(25,32)(27,31)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29138, #6463918, #6463925, … |
| maximal normal subgroup of | C2×#29138, #6463918, #6463925, … |
| maximal quotient of | #361893, #363832, #365265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22126);