Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2)(3,4)(5,6)(7,15,18,30)(8,12)(9,22,24,13)(10,14)(11,17)(16,31,19,33)(20,35,36,28)(21,29)(23,26)(25,32)(27,34);(5,7,10,18)(6,11,14,26)(8,19,21,16)(9,23,24,17)(12,27,29,25)(13,30,22,15)(20,32,36,34)(28,31,35,33);(2,4)(5,8,10,21)(6,12,14,29)(7,16,18,19)(9,20,24,36)(11,25,26,27)(13,28,22,35)(15,31,30,33)(17,32,23,34)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,d^b=dh,f^b=fgh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94419, #94444, #133364, … |
| maximal normal subgroup of | #94419, #94444, #133364, … |
| maximal quotient of | #129220, #129252, #129343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5686);