Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(5,6,10,14)(7,11,17,24)(8,12,19,26)(9,20,21,13)(15,27,29,22)(16,30,31,23)(18,25,32,33)(28,35,36,34);(1,2)(3,4)(5,7,10,17)(6,11,14,24)(8,15,19,29)(9,16,21,31)(12,22,26,27)(13,23,20,30)(18,28,32,36)(25,34,33,35);(2,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,25)(14,26)(16,28)(17,29)(20,33)(21,32)(23,34)(24,27)(30,35)(31,36);(5,9)(6,13)(7,16)(8,18)(10,21)(11,23)(12,25)(14,20)(15,28)(17,31)(19,32)(22,34)(24,30)(26,33)(27,35)(29,36)Polycyclic presentation:
2^8|a^2=h,b^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^b=fh,e^c=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139756, #139758, #139807, … |
| maximal normal subgroup of | #139756, #139758, #139807, … |
| maximal quotient of | #245248, #245250, #245253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16893);