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 | |
241 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2,4,6)(3,7,8,5)(9,10,12,15)(11,19,21,33)(13,26,24,39)(14,27,25,18)(16,30,28,22)(17,32,29,20)(23,38,37,35)(31,36,34,40);(1,3,4,8)(2,5,6,7)(10,15)(16,28)(17,29)(18,27)(19,33)(26,39)(31,34)(35,38);(9,11,20,36,14,23,24,30,12,21,32,40,25,37,13,22)(10,16,26,35,18,31,29,33,15,28,39,38,27,34,17,19);(11,21)(16,28)(19,33)(22,30)(23,37)(31,34)(35,38)(36,40)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=fg,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137530, #141279, #143372, … |
| maximal normal subgroup of | #137530, #141279, #143372, … |
| maximal quotient of | #494161, #495029, #495374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26720);