Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| 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 | |
254 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 254 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)(5,6,9,14,26,28,24,13,11,16,23,33,10,15,8,22)(7,17,19,12,27,29,18,34,21,35,36,31,20,32,25,30);(5,7,10,27,11,21,26,20)(6,12,28,30,16,31,15,34)(8,19,23,18,24,36,9,25)(13,32,33,35,22,29,14,17);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131281, #131282, #131286, … |
| maximal normal subgroup of | #131281, #131282, #131286, … |
| maximal quotient of | #136611, #136612, #136639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5991);