Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 77 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)(5,6,9,14,11,15,10,26)(7,16,18,33,19,32,25,34)(8,20,22,28,24,36,23,13)(12,27,29,17,31,35,30,21);(5,7,8,21,10,18,23,27,11,19,24,17,9,25,22,35)(6,12,28,16,14,30,36,34,15,31,13,32,26,29,20,33);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130208, #130209, #130243, … |
| maximal normal subgroup of | #130208, #130209, #130243, … |
| maximal quotient of | #137611, #137613, #137651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6054);