Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 793 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(2,3)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,20)(18,19);(1,2)(3,4)(8,17)(10,19)(11,16)(15,18);(1,3,4,2)(5,7)(6,10)(8,18)(9,14)(11,20)(12,19)(13,16)(15,17);(7,14)(10,19)(13,20)(15,18)Polycyclic presentation:
2^8|c^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394514, #394515, #394553, … |
| maximal normal subgroup of | #394514, #394515, #394553, … |
| maximal quotient of | #238563, #238566, #238570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16481);