Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 263 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:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,19,20,17)(14,18);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,19)(16,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19);(7,14)(10,18)(13,20)(17,19)Polycyclic presentation:
2^8|a^2=g,b^2=e,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #148009, #148011, #214704, … |
| maximal normal subgroup of | #148009, #148011, #214704, … |
| maximal quotient of | #188273, #188274, #188277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13621);