Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| 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 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #389 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 309 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:
(1,2)(3,4)(5,6,10,14,11,16,9,15)(7,17,20,28,21,31,19,12)(8,22,25,30,26,32,24,13)(18,34,35,29,36,27,23,33);(5,7,9,19,11,21,10,20)(6,12,14,17,16,28,15,31)(8,23,24,36,26,35,25,18)(13,29,22,27,30,33,32,34);(2,4)(5,8,9,24,11,26,10,25)(6,13,14,22,16,30,15,32)(7,18,19,23,21,36,20,35)(12,27,17,33,28,34,31,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11250, #358106, #358752, … |
| maximal normal subgroup of | C2×#11250, #358106, #358752, … |
| maximal quotient of | #120176, #120202, #120203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4238);