Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1693 conjugacy classes of subgroups; only the 563 normal subgroups are shown.
This group has 1693 conjugacy classes of subgroups — too many to draw. Only its 563 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 — 5 generators on 20 points:
(2,4)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16);(1,2,3,4);(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20);(7,14)(12,19)(15,18)(16,20)Polycyclic presentation:
2^8|b^2=fh,c^2=g,f^2=h,b^a=bf,c^a=cg,f^a=fh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60647⋊C3 |
| maximal subgroup of | #129999, #130003, #152056, … |
| maximal normal subgroup of | #129999, #130003, #152056, … |
| maximal quotient of | #6357190, #6357194, #6357196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53635);