Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×D16 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(2,6)(7,8)(9,10,15,19)(11,20,21,37)(12,17,14,29)(13,25,28,39)(16,31,32,23)(18,24,36,27)(22,38,30,35)(26,33,40,34);(1,2,5,6)(3,7,4,8)(11,21)(16,32)(20,37)(22,30)(23,31)(26,40)(33,34)(35,38);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,22)(13,26)(14,30)(15,21)(17,33)(18,35)(19,32)(20,25)(23,27)(24,31)(28,40)(29,34)(36,38)(37,39);(1,4,5,3)(2,7,6,8)(9,12,15,14)(10,17,19,29)(11,22,21,30)(13,24,28,27)(16,33,32,34)(18,25,36,39)(20,38,37,35)(23,26,31,40)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=e,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151539, #151541, #151591, … |
| maximal normal subgroup of | #151539, #151541, #151591, … |
| maximal quotient of | #497672, #497940, #498056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26957);