Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| 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 | #525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 177 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(3,8)(5,7)(10,16)(11,19)(17,28)(18,29)(20,31)(21,32)(30,38)(33,39);(2,6)(5,7)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=de,c^b=ceh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #135957, #136830, #137920, … |
| maximal normal subgroup of | #135957, #136830, #137920, … |
| maximal quotient of | #493267, #493270, #493283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26480);