Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,23)(10,13)(11,15)(12,18)(14,25)(17,29)(19,33)(20,26)(22,28)(24,35)(27,34)(30,31)(32,36);(5,7,9,19,10,20,24,34,11,17,25,31,8,18,22,32)(6,12,14,27,13,16,28,30,15,26,23,36,21,29,35,33);(1,3,4,2)(7,17)(12,26)(16,29)(18,20)(19,31)(27,36)(30,33)(32,34)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,f^2=g,g^2=h,b^a=bd,d^a=dh,f^a=fh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #128258, #128262, #128270, … |
| maximal normal subgroup of | #128258, #128262, #128270, … |
| maximal quotient of | #128244, #128254, #128276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5592);