Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C2×#534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1063 conjugacy classes of subgroups — too many to draw. Only its 239 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)(13,17,19,20)(14,18);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,20)(16,19);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20);(6,12)(10,18)(11,15)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^b=fh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #148467, #359528, #359549, … |
| maximal normal subgroup of | #148467, #359528, #359549, … |
| maximal quotient of | #193865, #193871, #193877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13948);