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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 107 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)(9,10,14,18)(11,19,20,33)(12,24,26,39)(13,27,25,17)(15,28,29,22)(16,31,32,21)(23,38,36,35)(30,37,34,40);(2,5)(6,8)(11,20)(15,29)(19,33)(22,28)(23,36)(30,34)(35,38)(37,40);(9,11,21,37,13,23,26,28,14,20,31,40,25,36,12,22)(10,15,24,35,17,30,32,33,18,29,39,38,27,34,16,19);(1,3,4,7)(2,6,5,8)(11,20)(15,29)(19,33)(22,28)(23,36)(30,34)(35,38)(37,40)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137498, #137504, #141215, … |
| maximal normal subgroup of | #137498, #137504, #141215, … |
| maximal quotient of | #494200, #495309, #495554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26737);