Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #400 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 311 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:
(2,3)(5,6,10,14,11,15,9,25)(7,16,19,28,20,27,18,12)(8,21,23,30,24,36,22,13)(17,26,33,31,34,29,32,35);(1,2,4,3)(5,7,9,18,11,20,10,19)(6,12,14,16,15,28,25,27)(8,17,22,32,24,34,23,33)(13,29,21,35,30,26,36,31);(1,3,4,2)(5,8,9,22,11,24,10,23)(6,13,14,21,15,30,25,36)(7,17,18,32,20,34,19,33)(12,26,16,31,28,29,27,35)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131398, #131404, #131405, … |
| maximal normal subgroup of | #131398, #131404, #131405, … |
| maximal quotient of | #120169, #120187, #120188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4225);