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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
841 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 841 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)(7,12)(8,20)(9,14)(10,25)(11,16)(13,23)(15,18)(17,30)(19,29)(21,36)(22,32)(24,28)(26,31)(27,34)(33,35);(1,2)(3,4)(5,7,11,19)(6,12,16,29)(8,17,22,33)(9,10,24,18)(13,27,31,36)(14,25,28,15)(20,35,32,30)(21,26,34,23);(2,4)(5,8)(6,13)(7,17)(9,21)(10,26)(11,22)(12,27)(14,30)(15,32)(16,31)(18,23)(19,33)(20,25)(24,34)(28,35)(29,36);(5,9,11,24)(6,14,16,28)(7,18,19,10)(8,21,22,34)(12,15,29,25)(13,30,31,35)(17,23,33,26)(20,27,32,36)Polycyclic presentation:
2^8|b^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139738, #152711, #152713, … |
| maximal normal subgroup of | #139738, #152711, #152713, … |
| maximal quotient of | #245173, #245177, #245185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16905);