Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1877 conjugacy classes of subgroups — too many to draw. Only its 303 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)Polycyclic presentation:
2^8|a^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fh,g^b=gh,e^c=eh58 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13598);