Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 121 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)(9,10,13,18,15,20,14,19)(11,16,22,31,23,32,29,36)(12,24,26,35,28,34,27,17)(21,33,25,39,37,40,38,30);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,36)(20,32)(21,27)(24,39)(26,37)(28,38)(30,34)(35,40);(9,12,15,28)(10,17,20,35)(11,21,23,37)(13,27,14,26)(16,30,32,39)(18,34,19,24)(22,38,29,25)(31,40,36,33);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137258, #137267, #211078, … |
| maximal normal subgroup of | #137258, #137267, #211078, … |
| maximal quotient of | #243282, #243283, #243284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16705);