Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(14,16)(15,20)(17,18);(6,9)(10,12)(13,14,18,20,19,15,17,16);(1,2,3,4)(13,15,18,16,19,14,17,20);(5,7,8,11)(6,10,9,12)(13,16,17,15,19,20,18,14)Polycyclic presentation:
2^8|b^2=g,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #102175, #102186, #102191, … |
| maximal normal subgroup of | #102175, #102186, #102191, … |
| maximal quotient of | #352615, #352620, #352622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26229);