Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| 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×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 111 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)(9,10,14,19)(11,15,20,28)(12,16,25,32)(13,26,24,18)(17,33,31,23)(21,27,34,38)(22,37,35,30)(29,40,39,36);(2,5)(6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,29)(18,30)(19,28)(23,36)(24,35)(25,34)(26,37)(31,39)(32,38)(33,40);(1,3,4,7)(2,6,5,8)(10,16)(12,23)(13,24)(15,27)(17,18)(19,32)(21,36)(22,35)(25,33)(26,31)(28,38)(29,30)(34,40)(37,39);(11,20)(15,28)(21,34)(22,35)(27,38)(29,39)(30,37)(36,40)Polycyclic presentation:
2^8|a^2=h,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #503860, #503861, #503864, … |
| maximal normal subgroup of | #503860, #503861, #503864, … |
| maximal quotient of | #493903, #494243, #494448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26639);