Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 349 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,14,10,13,21,27)(7,11,17,23,16,24,29,34)(8,12,20,26,19,25,32,35)(15,22,28,33,18,30,31,36);(5,7)(6,11)(8,18,19,15)(9,17)(10,16)(12,22,25,30)(13,24)(14,23)(20,31,32,28)(21,29)(26,33,35,36)(27,34);(1,2,3,4)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,25)(14,26)(16,18)(17,28)(21,32)(23,33)(24,30)(27,35)(29,31)(34,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93966, #136134, #136135, … |
| maximal normal subgroup of | #93966, #136134, #136135, … |
| maximal quotient of | #121330, #121359, #121360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4333);