Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,24)(13,18)(14,19)(16,26)(17,20)(21,37)(22,30)(23,32)(25,29)(27,34)(28,35)(31,36)(33,38)(39,40);(1,3,5,4)(2,6,8,7)(9,11,20,36,14,22,13,23)(10,15,17,31,19,30,18,32)(12,25,28,21,27,39,26,38)(16,33,24,29,35,37,34,40);(9,12,14,27)(10,16,19,35)(11,21,22,38)(13,28,20,26)(15,29,30,40)(17,34,18,24)(23,39,36,25)(31,33,32,37);(1,4,5,3)(2,7,8,6)(11,22)(15,30)(21,38)(23,36)(25,39)(29,40)(31,32)(33,37)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #332408, #332411, #332418, … |
| maximal normal subgroup of | #332408, #332411, #332418, … |
| maximal quotient of | #318716, #318719, #318724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21288);