Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
589 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 589 conjugacy classes of subgroups — too many to draw. Only its 127 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)(9,10,15,19)(11,16,22,33)(12,23,25,36)(13,27,14,18)(17,29,35,26)(20,34,37,40)(21,32,28,31)(24,39,38,30);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,24)(13,21)(14,28)(15,22)(17,34)(18,31)(19,33)(20,26)(23,39)(25,38)(27,32)(29,37)(30,36)(35,40);(1,3,5,4)(2,7,8,6)(9,12,13,26,15,25,14,29)(10,17,18,23,19,35,27,36)(11,20,21,38,22,37,28,24)(16,30,31,34,33,39,32,40);(1,4,5,3)(2,7,8,6)(12,25)(17,35)(20,37)(23,36)(24,38)(26,29)(30,39)(34,40)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152774, #152777, #250395, … |
| maximal normal subgroup of | #152774, #152777, #250395, … |
| maximal quotient of | #303262, #303263, #303270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18867);