Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2003 conjugacy classes of subgroups; only the 307 normal subgroups are shown.
This group has 2003 conjugacy classes of subgroups — too many to draw. Only its 307 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:
(1,2)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)Polycyclic presentation:
2^8|a^2=g,d^2=h,b^a=be,c^a=cf,e^a=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105822, #105832, #106146, … |
| maximal normal subgroup of | #105822, #105832, #106146, … |
| maximal quotient of | #184738, #184742, #184745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13572);