Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
885 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 885 conjugacy classes of subgroups — too many to draw. Only its 143 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)(8,11)(9,12)(10,15)(14,18)(16,20)(17,19);(1,2,4,3)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,18,19,20);(5,8)(6,11)(7,14,15,16)(9,17)(10,18,13,20)(12,19);(1,3,4,2)(6,12)(10,13)(11,19)(18,20)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=eh,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245615, #245625, #245627, … |
| maximal normal subgroup of | #245615, #245625, #245627, … |
| maximal quotient of | #297243, #297244, #297245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18761);