Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 179 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)(3,7)(4,6)(5,8)(9,10,13,18,14,19,25,33,12,17,23,31,24,32,36,40)(11,16,21,29,22,30,35,39,20,15,26,27,34,28,37,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,27)(19,28)(23,26)(24,34)(25,35)(29,31)(30,32)(33,38)(36,37)(39,40);(3,8)(5,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,29)(19,30)(23,26)(24,34)(25,35)(27,31)(28,32)(33,39)(36,37)(38,40);(2,6)(5,7)(10,17)(11,20)(18,31)(19,32)(21,26)(22,34)(33,40)(35,37)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=be,d^a=de,c^b=ceh,d^b=dh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #137822, #137825, #137842, … |
| maximal normal subgroup of | #137822, #137825, #137842, … |
| maximal quotient of | #493269, #493285, #493308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26483);