Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 607 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,28)(19,36)(20,34)(21,31)(23,35)(26,30)(27,33)(29,38)(32,39)(37,40);(2,5)(6,8)(9,11,14,21)(10,15,18,31)(12,19,23,37)(13,26,28,20)(16,32,33,29)(17,34,25,30)(22,36,35,40)(24,39,27,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,35)(18,33)(20,39)(21,37)(22,25)(24,28)(26,38)(30,40)(31,32)(34,36);(2,5)(6,8)(12,23)(16,33)(19,37)(22,35)(24,27)(29,32)(36,40)(38,39)Polycyclic presentation:
2^8|b^2=h,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139841, #144401, #502693, … |
| maximal normal subgroup of | #139841, #144401, #502693, … |
| maximal quotient of | #254601, #254608, #254609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17011);