Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(5,6,9,13,10,14,24,30)(7,15,18,32,16,31,22,35)(8,19,20,29,21,12,23,28)(11,25,27,33,26,34,36,17);(1,2,3,4)(5,7)(6,11,14,26)(8,17)(9,22)(10,16)(12,15,19,31)(13,27,30,36)(18,24)(20,34)(21,33)(23,25)(28,32,29,35);(7,16)(11,26)(15,31)(17,33)(18,22)(25,34)(27,36)(32,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=h,e^2=h,b^a=bd,d^a=dg,f^a=fh,g^a=gh,c^b=ch,e^b=egh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #146481, #146482, #146501, … |
| maximal normal subgroup of | #146481, #146482, #146501, … |
| maximal quotient of | #126915, #126954, #126979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5574);