Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 920 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,12)(7,14)(8,17)(9,10)(16,18)(19,20);(1,2)(3,4)(5,6)(7,15)(8,11)(9,17)(10,12)(13,16)(14,19)(18,20);(2,4)(5,7,11,16)(6,13,8,15)(9,14,10,18)(12,19,17,20);(5,8)(6,11)(7,15)(9,12)(10,17)(13,16)(14,19)(18,20)Polycyclic presentation:
2^8|c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,f^c=fh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #273324, #273326, #273360, … |
| maximal normal subgroup of | #273324, #273326, #273360, … |
| maximal quotient of | #244659, #244669, #244671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16867);