Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 973 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(5,6,11,16)(7,17,19,29)(8,21,23,30)(9,15,10,14)(12,24,27,20)(13,25,28,22)(18,26,32,36)(31,33,35,34);(1,2)(3,4)(5,7)(6,12)(8,18)(9,24)(10,20)(11,19)(13,26)(14,29)(15,17)(16,27)(21,35)(22,34)(23,32)(25,33)(28,36)(30,31);(2,4)(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,21)(15,30)(16,28)(17,31)(19,32)(20,33)(24,34)(27,36)(29,35);(7,19)(12,27)(17,29)(18,32)(20,24)(26,36)(31,35)(33,34)Polycyclic presentation:
2^8|a^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,d^b=dh,e^b=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142229, #142231, #142421, … |
| maximal normal subgroup of | #142229, #142231, #142421, … |
| maximal quotient of | #232216, #232218, #232222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16396);