Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| 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 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,20,27)(8,13)(9,14)(10,15)(11,18,12,19)(17,29,36,34)(21,28,35,26)(22,30)(23,31)(24,32)(25,33);(1,2)(3,4)(5,7)(6,11)(8,21)(9,19)(10,20)(12,15)(13,29)(14,27)(16,33)(17,22)(18,25)(23,35)(24,36)(26,30)(28,32)(31,34);(2,4)(5,7)(6,12)(8,17)(9,18)(10,20)(11,15)(13,28)(14,27)(16,33)(19,25)(21,22)(23,36)(24,35)(26,31)(29,32)(30,34);(5,8,10,23)(6,13,15,31)(7,17,20,36)(9,24,25,22)(11,26,12,28)(14,32,33,30)(16,34,27,29)(18,35,19,21)Polycyclic presentation:
2^8|a^2=efh,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125643, #125889, #126296, … |
| maximal normal subgroup of | #125643, #125889, #126296, … |
| maximal quotient of | #343029, #343033, #343037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24063);