Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 119 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)(3,6)(4,5)(7,8)(9,10)(11,24)(12,17)(13,19)(14,33)(15,35)(16,23)(18,29)(20,25)(21,34)(22,32)(26,40)(27,36)(28,37)(30,39)(31,38);(2,5)(6,8)(9,11,25,30,13,28,34,31,16,29,15,27,32,26,14,12)(10,17,33,40,22,36,35,18,23,38,21,37,19,39,20,24);(1,3,4,7)(2,6,5,8)(9,12,14,26,32,27,15,29,16,31,34,28,13,30,25,11)(10,18,20,36,19,40,21,17,23,24,35,39,22,37,33,38);(9,13,16,32)(10,19,23,22)(11,26,29,28)(12,27,31,30)(14,25,34,15)(17,36,38,39)(18,37,24,40)(20,21,35,33)Polycyclic presentation:
2^8|b^2=fh,c^2=e,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137575, #141318, #143271, … |
| maximal normal subgroup of | #137575, #141318, #143271, … |
| maximal quotient of | #494088, #494914, #495122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26696);