Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 8192 |
| 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 | |
289 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 111 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)(5,6,10,14,11,16,9,15)(7,17,19,12,21,28,20,29)(8,22,24,32,25,31,26,13)(18,30,35,36,23,27,34,33);(1,2,4,3)(5,7,9,20,11,21,10,19)(6,12,15,17,16,29,14,28)(8,23,24,34,25,18,26,35)(13,30,22,36,32,27,31,33);(1,3,4,2)(5,8,9,24,11,25,10,26)(6,13,14,31,16,32,15,22)(7,18,19,34,21,23,20,35)(12,27,17,33,29,30,28,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11264, #358278, #358423, … |
| maximal normal subgroup of | C2×#11264, #358278, #358423, … |
| maximal quotient of | #120694, #120710, #120714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4252);