Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 481 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)(5,6,9,13,10,14,25,32)(7,15,18,33,16,29,23,35)(8,19,21,31,22,12,24,30)(11,26,28,20,27,34,36,17);(5,7)(6,11)(8,20)(9,23)(10,16)(12,29)(13,28)(14,27)(15,19)(17,22)(18,25)(21,26)(24,34)(30,35)(31,33)(32,36);(2,4)(7,16)(11,27)(15,29)(17,20)(18,23)(26,34)(28,36)(33,35)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^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 | #94422, #94426, #94443, … |
| maximal normal subgroup of | #94422, #94426, #94443, … |
| maximal quotient of | #129218, #129251, #129341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5690);