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 | #397 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 297 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:
(1,2,3,4)(5,6)(7,16)(8,20)(9,25)(10,14)(11,15)(12,18)(13,23)(17,30)(19,28)(21,35)(22,31)(24,34)(26,29)(27,36)(32,33);(5,7)(6,12)(8,21,23,36)(9,26)(10,18)(11,19)(13,30,20,32)(14,28)(15,29)(16,25)(17,24,33,22)(27,31,35,34);(5,8,9,22,11,23,10,24)(6,13,14,31,15,20,25,34)(7,17,18,21,19,33,26,36)(12,27,16,32,29,35,28,30)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11281, #358296, #358952, … |
| maximal normal subgroup of | C2×#11281, #358296, #358952, … |
| maximal quotient of | #120669, #120670, #120734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4269);